Boundary singularities of positive solutions of quasilinear Hamilton–Jacobi equations

Abstract

We study the boundary behaviour of the solutions of (E) $-\Delta _p u+|\nabla u|^q=0$ in a domain $\Omega \subset \mathbb {R}^N$, when $N\ge p> q>p-1$. We show the existence of a critical exponent $q_*<p$ such that if $p-1<q<q_*$ there exist positive solutions of (E) with an isolated singularity on $\partial \Omega$ and that these solutions belong to two different classes of singular solutions. If $q_*\le q<p$ no such solution exists and actually any boundary isolated singularity of a positive solution of (E) is removable. We prove that all the singular positive solutions are classified according the two types of singular solutions that we have constructed.

1 Introduction

Let $N\ge p>1$, $q>p-1$ and $\Omega \subset \mathbb {R}^N$ ($N>1$) be a $C^2$ bounded domain such that $0\in \partial \Omega$. In this article we study the boundary behavior at 0 of nonnegative functions $u\in C^1(\Omega )\cap C(\overline{\Omega }{\setminus }\{0\})$ which satisfy

$$\begin{aligned} \left. \begin{array}{ll} -\Delta _p u+|\nabla u|^q=0&\quad \text {in }\Omega , \end{array}\right. \end{aligned}$$

(1.1)

where $\Delta _pu:=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$.where $\Delta _pu:=\mathrm{div}(|\nabla u|^{p-2}\nabla u)$. The two main questions we consider are as follows:

Q-1:

Existence of positive solutions of (1.1).

Q-2:

Description of positive solutions with an isolated boundary singularity at 0.

When $p=2$ a fairly complete description of positive solutions of

$$\begin{aligned} \left. \begin{array}{ll} -\Delta u+|\nabla u|^q=0\\ \end{array} \right. \end{aligned}$$

(1.2)

in $\Omega$ is provided by Nguyen-Phuoc and Véron [11]. In particular they prove the following series of results in the range of values $1<q<2$.

• 1. Any signed solution of (1.3) verifies the estimates

  $$\begin{aligned} |\nabla u(x)|\le c_{N,q}(d(x))^{-\frac{1}{q-1}}\qquad \forall x\in \Omega , \end{aligned}$$

  (1.3)

  where $d(x)=\text{ dist }\,(x,\partial \Omega )$. As a consequence, if $u\in C(\overline{\Omega }{\setminus }\{0\})$ is a solution which vanishes on $\partial \Omega {\setminus }\{0\}$, it satisfies

  $$\begin{aligned} |u(x)|\le c_{q,\Omega }d(x)|x|^{-\frac{1}{q-1}}\quad \forall x\in \Omega . \end{aligned}$$

  (1.4)

• 2. If $\frac{N+1}{N}\le q<2$ any positive solution of (1.3) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$ is identically 0. An isolated boundary point is a removable singularity for (1.2).

• 3. If $1<q<\frac{N+1}{N}$ and $k>0$ there exists a unique positive solution $u:=u_{k}$ of (1.2) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies $u(x)\sim c_N kP^\Omega (x,0)$ as $x\rightarrow 0$, where $P^\Omega$ is the Poisson kernel in $\Omega \times \partial \Omega$.

• 4. If $1<q<\frac{N+1}{N}$ there exists a unique positive solution u of (1.2) in the half-space $\mathbb {R}^N_+:=\{x=(x',x_N):x'\in \mathbb {R}^{N-1}, x_N>0\}$ under the form $u(x)=|x|^{-\frac{2-q}{q-1}}\omega (|x|^{-1}x)$ which vanishes on $\partial \mathbb {R}^N_+{\setminus }\{0\}$. The function $\omega$ is the unique positive solution of

  $$\begin{aligned}&\displaystyle -\Delta ' \omega +\left( \left( \frac{2-q}{q-1}\right) ^2\omega ^2+|\nabla '\omega |^2\right) ^{\frac{q}{2}}-\lambda _{N,q}\omega =0\quad \text {in }S^{N-1}_+,\nonumber \\&\displaystyle \omega =0\quad \text {in }\partial S^{N-1}_+, \end{aligned}$$

  (1.5)

  where $S^{N-1}$ is the unit sphere of $\mathbb {R}^N$, $\partial S^{N-1}_+=\partial \mathbb {R}^N_+\cap S^{N-1}$, $\Delta '$ the Laplace–Beltrami operator and $\lambda _{N,q}>0$ an explicit constant.

• 5. If $1<q<\frac{N+1}{N}$ and u is a positive solution of (1.3) in $\Omega$, which is continuous in $\overline{\Omega }{\setminus }\{0\}$ and vanishes on $\partial \Omega {\setminus }\{0\}$ the following dichotomy occurs:

• (i) either $u(x)\sim |x|^{-\frac{2-q}{q-1}}\omega (|x|^{-1}x)$ as $x\rightarrow 0$,

• (ii) or $u(x)\sim kc_NP^{\Omega }(x,0)$ as $x\rightarrow 0$ for some $k\ge 0$.

The aim of this article is to extend to the quasilinear case $1<p\le N$ the above mentioned results. The following pointwise gradient estimate valid for any signed solution u of (1.1) has been proved in [3]: if $0<p-1<q$ there exists a constant $c_{N,p,q}>0$ such that

$$\begin{aligned} \left| \nabla u(x)\right| \le c_{N,p,q}(d(x))^{-\frac{1}{q+1-p}}\quad \forall x\in \Omega . \end{aligned}$$

(1.6)

As a consequence, any solution $u\in C^1(\overline{\Omega }{\setminus }\{0\}$ satisfies

$$\begin{aligned} \left| u(x)\right| \le c_{p,q,\Omega }d(x)\left| x\right| ^{-\frac{1}{q+1-p}}\quad \forall x\in \Omega . \end{aligned}$$

(1.7)

Concerning boundary singularities, the situation is much more complicated than in the case $p=2$ and the threshold of critical exponent less explicit. We first consider the problem in $\mathbb {R}^N_+$. Assuming $p-1<q\le p$, separable solutions of (1.1) in $\mathbb {R}^N_+$ vanishing on $\partial \mathbb {R}^N_+{\setminus }\{0\}$ can be looked for in spherical coordinates $(r,\sigma )\in \mathbb {R}^*_+\times S^{N-1}$ (we denote $\mathbb {R}^*_+=(0,\infty )$) under the form

$$\begin{aligned} u(x)=u(r,\sigma )=r^{- \beta _q}\omega (\sigma ),\quad r>0,\;\sigma \in S^{N-1}_+:=\{S^{N-1}\cap \mathbb {R}^N_+\}. \end{aligned}$$

(1.8)

Then $\omega$ is solution of the following problem

$$\begin{aligned}&\displaystyle -div'\left( \left( \beta _q^2\omega ^2+|\nabla ' \omega |^2\right) ^{\frac{p-2}{2}}\nabla ' \omega \right) -\beta _q\Lambda _{\beta _q}\left( \beta _q^2\omega ^2+|\nabla ' \omega |^2\right) ^{\frac{p-2}{2}}\omega \nonumber \\&\displaystyle +\left( \beta _q^2\omega ^2+|\nabla ' \omega |^2\right) ^{\frac{q}{2}}=0\quad \text{ in } S_+^{N-1}\nonumber \\&\displaystyle \omega =0\quad \text{ on } \partial S_+^{N-1}, \end{aligned}$$

(1.9)

where

$$\begin{aligned} \beta _q=\frac{p-q}{q+1-p}\quad \text {and}\quad \Lambda _{\beta _q}=\beta _q(p-1)+p-N, \end{aligned}$$

(1.10)

and $\nabla '$ is the covariant derivative on $S^{N-1}$ identified to the tangential gradient thanks to the canonical isometrical imbedding of $S^{N-1}$ into $\mathbb {R}^N$, and $div'$ the divergence operator acting on vector fields on $S^{N-1}$.

The existence of a positive solution to this problem cannot be separated from the problem of existence of separable p -harmonic functions which are p-harmonic in $\mathbb {R}^N_+$ which vanish on $\partial \mathbb {R}^N_+{\setminus }\{0\}$ and have the form $\Psi (x)=\Psi (r,\sigma )=r^{-\beta }\psi (\sigma )$ for some real number $\beta$. Necessarily such a $\psi$ must satisfy

$$\begin{aligned}&\displaystyle -div'\left( \left( \beta ^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\nabla ' \psi \right) -\beta \Lambda _{\beta }\left( \beta ^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi =0\quad \text{ in } S_+^{N-1}\nonumber \\&\displaystyle \psi =0\quad \text{ on } \partial S_+^{N-1}, \end{aligned}$$

(1.11)

where $\Lambda _{\beta }=\beta (p-1)+p-N$. We will refer to (1.11) as the spherical p-harmonic eigenvalue problem. The study of this problem has been initiated in the 2-dim case by Krol [8] ($\beta <0$) and Kichenassamy and Véron [9] ($\beta >0$). In this case $\omega$ satisfies a completely integrable second order differential equation. In the case where $S^{N-1}_+$ is replaced by a smooth domain $S\subset S^{N-1}$ with $N\ge 3$, Tolksdorf [14] proved the existence of a unique couple $(\tilde{\beta }_s,\tilde{\psi }_s)$ where $\tilde{\beta }_s<0$ and $\tilde{\psi }_s$ has constant sign and is defined up to an homothety. Recently Porretta and Véron [12] gave a simpler and more general proof of the existence of two couples $(\tilde{\beta }_s,\tilde{\psi }_s)$ and $(\beta _{*\,s},\psi _{*\,s})$ where $\beta _{*\,s}>0$ and $\tilde{\psi }_s$ and $\psi _{*\,s}$ are positive solutions of (1.11) with $\beta =\tilde{\beta }_s$ and $\beta =\beta _{*\,s}$ respectively and are unique up to a multiplication by a real number. When $p=2$ this problem is an eigenvalue problem for the Laplace–Beltrami operator on a subdomain of $S^{N-1}$. If $S=S^{N-1}_+$, $\tilde{\beta }_s$ and $\beta _{*\,s}$ are respectively denoted by $\tilde{\beta }$ and $\beta _*$ and accordingly $\tilde{\psi }_s$ and $\psi _{*\,s}$ by $\tilde{\psi }$ and $\psi _*$. Since $x\mapsto x_N$ is p-harmonic, $\tilde{\beta }=-1$. Except in the cases $N=2$ where it is the positive root of some algebraic equation of degree 2, $p=2$ where it is $N-1$ and $p=N$ where it is 1, the value of $\beta _*$ is unknown besides the straightforward estimate $\beta _*\ge \max \{1,\frac{N-p}{p-1}\}$. Using the fact that $\psi _*$ depends only on the azimuthal variable and satisfies a differential equation, we prove in Appendix B the following new estimate:

Theorem A

Let $1<p\le N$.

• (i) If $2\le p\le N$, then $\beta _*\le \frac{N-1}{p-1}$ with equality only if $p=2$ or N.

• (ii) If $1\le p<2$, then $\beta _*> \frac{N-1}{p-1}$.

The p-harmonic function $\Psi _*(x)=\Psi _*(r,\sigma )=r^{-\beta _*}\psi _*(\sigma )$ endows the role of a Poisson kernel. To this exponent $\beta _*$ is associated the critical value $q_*$ of q defined by $\beta _*=\beta _q$, or equivalently

$$\begin{aligned} q_*:=\frac{\beta _*(p-1)+p}{\beta _*+1}=p-\frac{\beta _*}{\beta _*+1}. \end{aligned}$$

(1.12)

The following result characterizes strong singularities.

Theorem B

Let $0<p-1\le N$, then

• (i) If $p-1<q<q_*$ problem (1.9) admits a unique positive solution $\omega _*$.

• (ii) If $q_*\le q <p$ problem (1.9) admits no positive solution.

This critical exponent corresponds to the threshold of criticality for boundary isolated singularities.

Theorem C

Assume $q_*\le q <p\le N$. If $u\in C^1(\overline{\Omega }{\setminus }\{0\})$ is a nonnegative solution of (1.1) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$, it is identical zero.

As in the case $p=2$, there exist positive solutions (1.1) in $\Omega$ with weak boundary singularities which are characterized by their blow-up near the singularity. By opposition to the case $p=2$ where existence is obtained by use of a weak formulation of the boundary value problem, combined with uniform integrability of the absorption term thanks to Poisson kernel estimates (see [11]), this approach cannot be performed in the case $p\ne 2$; the obtention of solutions with weak singularities necessitates a very long and delicate construction of subsolutions and supersolutions. Furthermore, when $p\ne N$, the construction is done only if $\Omega$ is locally an hyperplane near 0.

In the sequel we denote by $B_R(a)$ the open ball of center a and radius $R>0$ and $B_R=B_R(0)$. We also set $B^+_R(a):=\mathbb {R}^N_+\cap B_R(a)$, $B^+_R:=\mathbb {R}^N_+\cap B_R$, $B^-_R(a):=\mathbb {R}^N_-\cap B_R(a)$ and $B^-_R:=\mathbb {R}^N_-\cap B_R$, where $\mathbb {R}^N_-:=\{x=(x',x_{N}):x'\in \mathbb {R}^{N-1}, x_N<0\}$. If $\Omega$ is an open domain and $R>0$, we put $\Omega _{R}=\Omega \cap B_{R}$.

Theorem D

Let $\Omega \subset \mathbb {R}^N_+$ be a bounded domain such that $0\in \partial \Omega$. Assume there exists $\delta >0$ such that $\Omega _\delta =B^+_\delta$ and $0<p-1<q<q_* <p\le N$. Then for any $k>0$ there exists a unique $u:=u_k\in C^1(\overline{\Omega }{\setminus }\{0\})$, solution of (1.1) in $\Omega$, vanishing on $\partial \Omega {\setminus }\{0\}$ and such that

$$\begin{aligned} \lim _{\begin{array}{c} {x\rightarrow 0}\\ \frac{x}{|x|}\rightarrow \sigma \in S^{N-1}_+ \end{array}}\!\!\!\!|x|^{\beta _*}u_k(x)=k\psi _*(\sigma ). \end{aligned}$$

(1.13)

Furthermore $\lim _{k\rightarrow \infty }u_{k}=u_\infty$ and

$$\begin{aligned} \lim _{\begin{array}{c} {x\rightarrow 0}\\ \frac{x}{|x|}\rightarrow \sigma \in S^{N-1}_+ \end{array}}\!\!\!\!|x|^{\beta _q}u_\infty (x)=\psi _*(\sigma ). \end{aligned}$$

(1.14)

When $p=N$, then $q_*=N-\frac{1}{2}$; in such a range of values we use the conformal invariance of $\Delta _N$ and prove that the previous result holds if $\Omega$ is any $C^2$ domain. Finally, the isolated singularities of positive solutions of (1.1) are completely described by the two types of singular solutions obtained in the previous theorem and we prove:

Theorem E

Let $\Omega$ be a bounded domain such that $0\in \partial \Omega$. Assume there exists $\delta >0$ such that $\Omega _\delta =B^+_\delta$ and $0<p-1<q<q_* <p\le N$. If $u\in C^1(\overline{\Omega }{\setminus }\{0\})$ is a positive solution of (1.1) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$, then

• (i) either there exists $k\ge 0$ such that

  $$\begin{aligned} \lim _{\begin{array}{c} {x\rightarrow 0}\\ \frac{x}{|x|}\rightarrow \sigma \in S^{N-1}_+ \end{array}}\!\!\!\!|x|^{\beta _*}u(x)=k\psi _*(\sigma ); \end{aligned}$$

  (1.15)

• (ii) or

  $$\begin{aligned} \lim _{\begin{array}{c} {x\rightarrow 0}\\ \frac{x}{|x|}\rightarrow \sigma \in S^{N-1}_+ \end{array}}\!\!\!\!|x|^{\beta _q}u(x)=\psi _*(\sigma ). \end{aligned}$$

  (1.16)

2 A priori estimates

2.1 The gradient estimates and its applications

We recall the following estimate and its consequences which are proved in [3].

Proposition 2.1

Assume $q>p-1$ and u is a $C^1$ solution of (1.1) in a domain $\Omega$. Then

$$\begin{aligned} \left| \nabla u(x)\right| \le c_{N,p,q}(d(x))^{-\frac{1}{q+1-p}}\quad \forall x\in \Omega . \end{aligned}$$

(2.1)

The first application is a pointwise upper bound for solutions with isolated singularities.

Corollary 2.2

Assume $q>p-1>0$, $R^*>0$ and $\Omega$ is a domain containing 0 such that $d(0)\ge 2R^*$. Then for any $x\in B_{R^*}{\setminus }\{0\}$, and $0<R\le R^*$, any $u\in C^1(\Omega {\setminus }\{0\})$ solution of (1.1) in $(\Omega {\setminus }\{0\})$ satisfies

$$\begin{aligned} \left| u(x)\right| \le c_{N,p,q}\left| |x|^{\frac{q-p}{q+1-p}}-R^{\frac{q-p}{q+1-p}}\right| +\max \{\left| u(z)\right| :\left| z\right| =R\}, \end{aligned}$$

(2.2)

if $p\ne q$, and

$$\begin{aligned} \left| u(x)\right| \le c_{N,p}\left( \ln R-\ln \left| x\right| \right) +\max \{\left| u(z)\right| :\left| z\right| =R\}, \end{aligned}$$

(2.3)

if $p=q$.

The second application corresponds to solutions with boundary blow-up. For $\delta >0$ small enough we set $\Omega _{\delta }:=\{z\in \Omega :d(z)<\delta \}$.

Corollary 2.3

Assume $q>p-1>0$, $\Omega$ is a bounded domain with a $C^2$ boundary. Then there exists $\delta _1>0$ which depends only on $\Omega$ such that any $u\in C^1(\Omega )$ solution of (1.1) in $\Omega$ satisfies

$$\begin{aligned} \left| u(x)\right| \le c_{N,p,q}\left| (d(x))^{\frac{q-p}{q+1-p}}-\delta _1^{{\frac{q-p}{q+1-p}}}\right| +\max \{\left| u(z)\right| :d(z)=\delta _1\}\quad \forall x\in \Omega _{\delta _1} \end{aligned}$$

(2.4)

if $p\ne q$, and

$$\begin{aligned} \left| u(x)\right| \le c_{N,p,q}\left( \ln \delta _1-\ln d(x)\right) +\max \{\left| u(z)\right| :d(z)=\delta _1\}\quad \forall x\in \Omega _{\delta _1} \end{aligned}$$

(2.5)

if $p=q$.

Remark

As a consequence of (2.4) there holds for $p>q>p-1$

$$\begin{aligned} u(x)\le \left( c_{N,p,q}+K\max \{\left| u(z)\right| :d(z)\ge \delta _1\}\right) (d(x))^{\frac{q-p}{q+1-p}}\quad \forall x\in \Omega \end{aligned}$$

(2.6)

where $K=(\mathrm{diam} (\Omega ))^{\frac{p-q}{q+1-p}}$, with the standard modification if $p=q$.

As a variant of Corollary 2.3 the following upper estimate of solutions in an exterior domain will be used in the sequel.

Corollary 2.4

Assume $q>p-1>0$, $R>0$ and $u\in C^1(B_{R_0}^c)$ is any solution of (1.1) in $B_{R_0}^c$. Then for any $R>R_0$ there holds

$$\begin{aligned} \left| u(x)\right| \le c_{N,p,q}\left| (\left| x\right| -R_0)^{\frac{q-p}{q+1-p}}-(R-R_0)^{\frac{q-p}{q+1-p}}\right| +\max \{\left| u(z)\right| :\left| z\right| =R\}\quad \forall x\in B_{R}^c\nonumber \\ \end{aligned}$$

(2.7)

if $p\ne q$ and

$$\begin{aligned} \left| u(x)\right| \le c_{N,p,q}\left( \ln (\left| x\right| -R_0)-\ln (R-R_0)\right) +\max \{\left| u(z)\right| :\left| z\right| =R\}\quad \forall x\in B_{R}^c\quad \end{aligned}$$

(2.8)

if $p=q$.

Proof

The proof is a consequence of the identity

$$\begin{aligned} u(x)=u(z)+{\int _{0}^{1}}{\frac{d}{dt} }u(tx+(1-t)z)dt={\int _{0}^{1}}\langle \nabla u(tx+(1-t)z),x-z\rangle dt \end{aligned}$$

where $z=\frac{R}{\left| x\right| }x$. Since by (2.1)

$$\begin{aligned} \left| \nabla u(tx+(1-t)z)\right| \le C_{N,p,q}(t\left| x\right| +(1-t)R-R_0)^{-\frac{1}{q+1-p}}, \end{aligned}$$

Equations (2.7) and (2.8) follow by integration. $\square$

2.2 Boundary a priori estimates

The next result is the extension to isolated boundary singularities of a previous regularity estimate dealing with singularity in a domain proved in [3, Lemma 3.10].

Lemma 2.5

Assume $p-1<q<p$, $\Omega$ is a bounded $C^2$ domain such that $0\in \partial \Omega$. Let $u\in C^1(\overline{\Omega }{\setminus }\{0\})$ be a solution of (1.1) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies

$$\begin{aligned} \left| u(x)\right| \le \phi (\left| x\right| )\quad \forall x\in \Omega , \end{aligned}$$

(2.9)

where $\phi :\mathbb {R}^*_+\mapsto \mathbb {R}_+$ is continuous, nonincreasing and satisfies

$$\begin{aligned} \phi (rs)\le \gamma \phi (r)\phi (s)\quad \text {and}\quad r^{\frac{p-q}{q+1-p}}\phi (r)\le c, \end{aligned}$$

(2.10)

for some $\gamma ,c>0$ and any $r,s>0$. There exist $\alpha \in (0,1)$ and $c_1=c_1(p,q,\Omega )>0$ such that

$$\begin{aligned} \begin{array}{lll} (i)&{}\left| \nabla u(x)\right| \le c_1\phi (\left| x\right| )\left| x\right| ^{-1}&{}\quad \forall x\in \Omega ,\\ (ii)&{}\left| \nabla u(x)-\nabla u(y)\right| \le c_1\phi (\left| x\right| )\left| x\right| ^{-1-\alpha }\left| x-y\right| ^\alpha &{}\quad \forall x,y\in \Omega ,\;\left| x\right| \le \left| y\right| . \end{array} \end{aligned}$$

(2.11)

Furthermore

$$\begin{aligned} u(x)\le c_1\phi (\left| x\right| )\frac{d(x)}{\left| x\right| }\quad \forall x\in \Omega . \end{aligned}$$

(2.12)

Proof

For $\ell >0$, we set $\Omega ^\ell :=\frac{1}{\ell }\Omega$. If $\ell \in (0,1]$ the curvature of $\partial \Omega ^\ell$ remains uniformly bounded. As in [5, p 622], there exists $0<\delta _0\le 1$ and an involutive diffeomorphism $\psi$ from $\overline{B}_{\delta _0}\cap \overline{\Omega }^{\delta _0}$ into $\overline{B}_{\delta _0}\cap (\Omega ^{\delta _0})^c$ which is the identity on $\overline{B}_{\delta _0}\cap \partial \Omega ^{\delta _0}$ and such that $D\psi (\xi )$ is the symmetry with respect to the tangent plane $T_\xi \partial \Omega$ for any $\xi \in \partial \Omega \cap \overline{B}_{\delta _0}$. We extend any function v defined in $\overline{B}_{\delta _0}\cap \overline{\Omega }^{\delta _0}$ and vanishing on $\overline{B}_{\delta _0}\cap \partial \Omega ^{\delta _0}$ into a function $\tilde{v}$ defined in $\overline{B}_{\delta _0}$ by

$$\begin{aligned} \tilde{v}(x)=\left\{ \begin{array} {ll} v(x)&{}\quad \text {if }x\in \overline{B}_{\delta _0}\cap \overline{\Omega }^{\delta _0}\\ -v\circ \psi (x)&{}\quad \text {if }x\in \overline{B}_{\delta _0}\cap (\Omega ^{\delta _0})^c, \end{array}\right. \end{aligned}$$

(2.13)

If $v\in C^1(\overline{B}_{\delta _0}\cap \overline{\Omega }^{\delta _0})$ is a solution of (1.1) in $B_{\delta _0}\cap \Omega ^{\delta _0}$ which vanishes on $\partial \Omega ^{\delta _0}\cap \overline{B}_{\delta _0}$, $\tilde{v}$ satisfies

$$\begin{aligned} -\sum _j{\frac{\partial }{\partial x_j} }\tilde{A}_j(x,\nabla \tilde{v})+B(x,\nabla \tilde{v})=0\quad \text {in }B_{\delta _0}. \end{aligned}$$

(2.14)

As in [5, (2.37)] the $A_j$ and B satisfy the following estimates

$$\begin{aligned} \begin{array}{ll} (i) &{}\quad \tilde{A}_j(x,0)=0\\ (ii)&{}\quad \displaystyle \sum _{i,j}{\frac{\partial }{\partial \eta _i} }\tilde{A}_j(x,\eta )\xi _i\xi _j\ge C_1\left| \eta \right| ^{p-1}\left| \xi \right| ^2\\ (iii)&{}\quad \displaystyle \sum _{i,j}\left| {\frac{\partial }{\partial \eta _j} }\tilde{A}_j(x,\eta )\right| \le C_2\left| \eta \right| ^{p-2}, \end{array} \end{aligned}$$

(2.15)

and

$$\begin{aligned} \left| B(x,\eta )\right| \le C_3(1+\left| \eta \right| )^p, \end{aligned}$$

(2.16)

where the $C_j$ are positive constants. These estimates are the ones needed to apply Tolksdorf’s result [15, Th1, 2]. There exists a constant C, such that for any ball $\overline{B}_{3R}\subset \overline{B}_{\delta _0}$, there holds

$$\begin{aligned} \left\| \nabla \tilde{v}\right\| _{L^\infty (B_R)}\le C, \end{aligned}$$

(2.17)

where C depends on the constants $C_k$ ($k=1,2,3$), N, p and $\left\| \tilde{v}\right\| _{L^\infty (B_{3R})}$. We define

$$\begin{aligned} \Phi _\ell [u](y):=u_\ell =\frac{1}{\phi (\ell )} u(\ell y)\quad \forall y\in \Omega ^\ell . \end{aligned}$$

(2.18)

Then

$$\begin{aligned} |u_\ell (y)|\le \frac{\phi (\ell \left| y\right| )}{\phi (\ell )}\le \gamma \phi (\left| y\right| )\quad \forall y\in \Omega ^\ell \end{aligned}$$

(2.19)

and

$$\begin{aligned} -\Delta _pu_\ell +(\ell ^{\beta _q}\phi (\ell ))^{q+1-p}\left| \nabla u_\ell \right| ^q=0\quad \text {in }\Omega ^\ell . \end{aligned}$$

(2.20)

Using formula (2.13) we extend $u_\ell$ into a function $\tilde{u}_\ell$ which satisfies

$$\begin{aligned} -\sum _j{\frac{\partial }{\partial y_j} }\tilde{A}_j(y,\nabla \tilde{u}_\ell )+(\ell ^{\beta _q}\phi (\ell ))^{q+1-p}B(y,\nabla \tilde{u}_\ell )=0\quad \text {in }B_{\delta _0}. \end{aligned}$$

(2.21)

For $0<\left| x\right| <\delta _0$ there exists $\ell \in (0,2)$ such that $\frac{\delta _0\ell }{2}\le \left| x\right| \le \delta _0\ell$. Then $y\mapsto \tilde{u}_\ell (y)$ with $y=\frac{x}{\ell }$ satisfies (2.21) in $B_{\delta _0}$ and $|\tilde{u}_\ell (y)|\le \gamma _*\phi (\left| y\right| )$ since $\psi$ is a diffeomorphism and $D\psi (\xi )\in O(N)$ for any $\xi \in \partial \Omega \cap B_{\delta _0}$. The function $\tilde{u}_\ell$ remains bounded on any ball $B_{3R}(z)\subset \Gamma :=\{y\in \mathbb {R}^N:\frac{\delta _0}{2}<\left| y\right| <\delta _0\}$, therefore $\left| \nabla \tilde{u}_\ell (y)\right| \le c$ for any $y\in B_{R}(z)$, for some constant $c>0$. This implies

$$\begin{aligned} \begin{array}{ll} \left| \nabla u(x)\right| \le c\gamma _*\delta _0\phi \left( \frac{2}{\delta _0}\right) \phi (|x|)|x|^{-1}\quad \forall x\in \Omega \cap B_{\delta _0}, \end{array} \end{aligned}$$

(2.22)

which is (2.11)-(i). Moreover, by standard regularity estimates [10], there exists $\alpha \in (0,1)$ such that $\left| \nabla \tilde{u}_\ell (y)-\nabla \tilde{u}_\ell (y')\right| \le c\left| y-y'\right| ^\alpha$ for all y and $y'$ belonging to $B_{R}(z)$. This implies (2.11)-(ii).

Next we prove (2.12). Let $0<\delta _1\le \delta _0$ such that at any boundary point z there exist two closed balls of radius $\delta _1$ tangent to $\partial \Omega$ at z and which are included in $\Omega \cup \{z\}$ and in $\overline{\Omega }^c\cup \{z\}$ respectively ($\delta _1$ corresponds to the maximal radius of the interior and exterior sphere condition). Let $x\in \Omega$ such that $d(x)\le \delta _1$ (this is not a loss of generality) and $z_x$ be the projection of x on $\partial \Omega$. We first assume that x does not belong to the cone $\Sigma _{\frac{\pi }{4}}$ with vertex 0, axis $-\mathbf{n}_0$, where $\mathbf{n}_0$ is the normal outward unit vector at 0, and angle $\frac{\pi }{4}$. Consider the path $\zeta$ from $z_x$ to x defined by $\zeta (t)=tx+(1-t)z_x$ with $0\le t\le 1$. Then

$$\begin{aligned} u(x)={\int _{0}^{1}}{\frac{d}{dt} }u\circ \zeta (t)dt={\int _{0}^{1}}\langle \nabla u\circ \zeta (t),x-z_x\rangle dt \end{aligned}$$

(2.23)

Thus, by the Cauchy–Schwarz inequality, using (2.11),

$$\begin{aligned} \left| u(x)\right| \le c_1d(x){\int _{0}^{1}}{\frac{\phi (|\zeta (t)|)}{\left| \zeta (t)\right| } }dt. \end{aligned}$$

(2.24)

Since $x\notin \Sigma _{\frac{\pi }{4}}$, $\zeta (t)\notin \Sigma _{\frac{\pi }{4}}$ and there exists $c_2>0$ depending on $\Omega$ such that $c^{-1}_2\left| x\right| \le \left| \zeta (t)\right| \le c_2\left| x\right|$ for all $0\le t\le 1$. Therefore $\phi (|\zeta (t)|)\le \phi (c_2\left| x\right| )\le \gamma \phi (c_2)\phi (\left| x\right| )$ by (2.10). This implies

$$\begin{aligned} \left| u(x)\right| \le \gamma c_1c_2\phi (c_2){\frac{d(x)\phi (\left| x\right| )}{\left| x\right| } } \end{aligned}$$

(2.25)

by (2.12) whenever $x\notin \Sigma _{\frac{\pi }{4}}$. When $x\in \Sigma _{\frac{\pi }{4}}$ then $d(x)\le \left| x\right| \le c_3d(x)$ where $c_3>0$ depends on the curvature of $\partial \Omega$. Then (2.9) combined with (2.10) implies the claim. $\square$

Lemma 2.6

Assume $p-1<q\le p$, $\Omega$ is a bounded $C^2$ domain such that $0\in \partial \Omega$ and $R_0=\max \{ \left| z\right| :z\in \Omega \}$. If $u\in C(\overline{\Omega }{\setminus }\{0\})\cap C^1(\Omega )$ is a positive solution of (1.1) which vanishes on $\partial \Omega {\setminus }\{0\}$, it satisfies

$$\begin{aligned} u(x)\le \left\{ \begin{array}{ll} c_2\left( \left| x\right| ^{\frac{q-p}{q+1-p}}-R_0^{\frac{q-p}{q+1-p}}\right) &{}\quad \text {if }q< p\\ (p-1)\ln \left( \frac{R_0}{\left| x\right| }\right) &{}\quad \text {if }q= p \end{array}\right. \end{aligned}$$

(2.26)

for all $x\in \Omega$, where $c_2=c_2(p,q)>0$.

Proof

For $\epsilon >0$ we denote by $P_\epsilon :\mathbb {R}\mapsto \mathbb {R}_+$ the function defined by

$$\begin{aligned} P_\epsilon (r)=\left\{ \begin{array}{ll}0&{}\quad \text {if }0\le r\le \epsilon \\ -\frac{r^4}{2\epsilon ^3}+\frac{3r^3}{\epsilon ^2}-\frac{6r^2}{\epsilon }+5r-\frac{3\epsilon }{2}&{}\quad \text {if }\epsilon <r<2 \epsilon \\ r-\frac{3\epsilon }{2}&{}\quad \text {if }r\ge 2\epsilon , \end{array}\right. \end{aligned}$$

(2.27)

and by $u_\epsilon$ the extension of $P_\epsilon (u)$ by zero outside $\Omega$. There exists $R_0$ such that $\Omega \subset B_{R_0}$. Since $0\le P_\epsilon (r)\le |r|$ and $P_\epsilon$ is convex, $u_\epsilon \in C(\mathbb {R}^N{\setminus }\{0\})\cap W^{1,p}_{loc}(\mathbb {R}^N{\setminus }\{0\})$ and

$$\begin{aligned} -\Delta _pu_\epsilon +\left| \nabla u_\epsilon \right| ^q\le 0\qquad \text {in }\mathbb {R}^N. \end{aligned}$$

Let $R>R_0$. If $p-1<q<p$

$$\begin{aligned} U_{\epsilon ,R}(\left| x\right| )=c_2 \left( (|x|-\epsilon )^{\frac{q-p}{q+1-p}}-(R-\epsilon )^{\frac{q-p}{q+1-p}}\right) \quad \text {in }B_R{\setminus } B_\epsilon , \end{aligned}$$

(2.28)

with $c_2=(p-q)^{-1}(q+p-1)^{\frac{q-p}{q+1-p}}$. Then $-\Delta _pU_{\epsilon ,R}+\left| \nabla U_{\epsilon ,R}\right| ^q\ge 0$. Since $u_\epsilon$ vanishes on $\partial B_R$ and is finite on $\partial B_\epsilon$, it follows $u_\epsilon \le U_{\epsilon ,R}$. Letting successively $\epsilon \rightarrow 0$ and $R\rightarrow R_0$ yields to (2.26). If $q=p$ we take

$$\begin{aligned} U_{\epsilon ,R}(\left| x\right| )=(p-1) \ln \left( \frac{R-\epsilon }{\left| x\right| -\epsilon }\right) \quad \text {in }B_R{\setminus } B_\epsilon , \end{aligned}$$

(2.29)

which turns out to be a supersolution of (1.1); the end of the proof is similar.

As a consequence of Lemmas 2.5 and 2.6, we obtain. $\square$

Corollary 2.7

Let p, q $\Omega$ and u be as in Lemma 2.6. Then there exists a constant $c_3=c_3(p,q,\Omega )>0$ such that

$$\begin{aligned} \left| \nabla u(x)\right| \le c_3\left| x\right| ^{-\frac{1}{q+1-p}}\quad \forall x\in \Omega \end{aligned}$$

(2.30)

and

$$\begin{aligned} u(x)\le c_3d(x)\left| x\right| ^{-\frac{1}{q+1-p}}\quad \forall x\in \overline{\Omega }{\setminus }\{0\}. \end{aligned}$$

(2.31)

Remark

If $\Omega$ is locally flat near 0, then estimates (2.30) and (2.31) are valid without any sign assumption on u. More precisely, if $\partial \Omega \cap B_{\delta _0}=T_0\partial \Omega \cap B_{\delta _0}$ we can perform the reflection of u through the tangent plane $T_0\partial \Omega$ to $\partial \Omega$ at 0 and the new function $\tilde{u}$ is a solution of (1.1) in $B_{\delta _0}{\setminus }\{0\}$. By Proposition 2.1, it satisfies

$$\begin{aligned} \left| \nabla \tilde{u}(x)\right| \le c_{N,p,q}\left| x\right| ^{-\frac{1}{q+1-p}}\quad \forall x\in B_{\frac{\delta _0}{2}}{\setminus }\{0\}. \end{aligned}$$

(2.32)

Integrating this relation as in [3], we derive that for any $x\in B_{\frac{\delta _0}{2}}\cap \Omega$, there holds

$$\begin{aligned} \left| u(x)\right| \le \left\{ \begin{array}{ll} c_{N,p,q}\left( |x|^{-\beta _q}-\left( \frac{\delta _0}{2}\right) ^{-\beta _q}\right) +\max \{\left| u(z)\right| :\left| z\right| =\frac{\delta _0}{2}\}&{}\quad \text {if } p\ne q\\ c_{N,p}\ln \left( \frac{\delta _0}{2\left| x\right| }\right) +\max \{\left| u(z)\right| :\left| z\right| =\frac{\delta _0}{2}\}&{}\quad \text {if } p= q. \end{array}\right. \end{aligned}$$

(2.33)

In the next result we allow the boundary singular set to be a compact set.

Proposition 2.8

Let $p-1<q<p$ and $\delta _1$ as above. There exist $r^*\in (0, \delta _1]$ and $c_4=c_4(N,p,q)>0$ such that for any nonempty compact set $K\subset \partial \Omega$, $K\ne \partial \Omega$ and any positive solution $u\in C(\overline{\Omega }{\setminus } K)\cap C^1(\Omega )$ of (1.1) which vanishes on $\partial \Omega {\setminus } K$, there holds

$$\begin{aligned} u(x)\le c_4d(x)(d_K(x))^{-\frac{1}{q+1-p}}\quad \forall x\in \partial \Omega \;\text {s.t.}\;d(x)\le r^*, \end{aligned}$$

(2.34)

where $d_K(x)=\text{ dist }\,(x,K)$.

Proof

Step 1: tangential estimates Let $x\in \Omega$ such that $d(x)\le \delta _1$. We denote by $\sigma (x)$ the projection of x onto $\partial \Omega$, unique since $d(x)\le \delta _1$. Let $r\,,r',\tau >0$ such that $\frac{3}{4}r<r'<\frac{7}{8}r$ and $0<\tau \le \frac{r'}{2}$ and put $\omega _{\tau ,x}=\sigma (x)+\tau \mathbf{n}_{\sigma (x)}$. Since $\partial \Omega$ is $C^2$, there exists $0< r^*\le \delta _1$ depending on $\Omega$ such that $d_K(\omega _{\tau ,x})>\frac{7}{8}r$ whenever $d(x)\le r^*$. Let $a>0$ and $b>0$ to be specified later on; we define $\tilde{v}(s)=a(r'-s)^{\frac{q-p}{q+1-p}}-b$ and $v(y)=\tilde{v}(\left| y-\omega _{\tau ,x}\right| )$ in $[0,r')$ and $B_{r'}(\omega _{\tau ,x})$ respectively. Then

$$\begin{aligned} \left| \tilde{v}'\right| ^{p-2}\left( \left| \tilde{v}'\right| ^{q+2-p}-(p-1)\tilde{v}''-{\frac{N-1}{s} }\tilde{v}'\right) \!=\! a^{p-1}\left( {\frac{p-q}{q+1-p} }\right) ^{p-1} (r'-s)^{-\frac{q}{q+1-p}}X(s) \end{aligned}$$

where

$$\begin{aligned} X(s)=\left( a{\frac{p-q}{q+1-p} }\right) ^{q+1-p}-{\frac{p-1}{q+1-p} }-{\frac{(N-1)(r'-s)}{s} }. \end{aligned}$$

For any $\tau \in (0,r')$ there exists $a>0$ such that

$$\begin{aligned} \left( a{\frac{p-q}{q+1-p} }\right) ^{q+1-p}\ge {\frac{p-1}{q+1-p} }+{\frac{(N-1)(r'-s)}{s} }\quad \forall \tau \le s\le r'. \end{aligned}$$

This implies

$$\begin{aligned} -\Delta _p v+\left| \nabla v\right| ^q\ge 0\quad \text {in }B_{r'}(\omega _{\tau ,x}){\setminus } B_{\tau }(\omega _{\tau ,x}). \end{aligned}$$

(2.35)

Next we take $b=a(r'-\tau )^{\frac{q-p}{q+1-p}}$, thus $v=0$ on $\partial B_{\tau }(\omega _{\tau ,x})$. Clearly $B_{\tau }(\omega _{\tau ,x})\subset \overline{\Omega }^c$ since $\tau <\delta _1$. Therefore $v\ge 0=u$ on $\partial \Omega \cap B_{r'}(\omega _{\tau ,x})$ and $u\le v=\infty$ on $\Omega \cap \partial B_{r'}(\omega _{\tau ,x})$. By the comparison principle, $v\ge u$ in $\Omega \cap B_{r'}(\omega _{\tau ,x}).$ In particular

$$\begin{aligned} u(x)\le v(x)\le a(r'-\tau -d(x))^{\frac{q-p}{q+1-p}}-a(r'-\tau )^{\frac{q-p}{q+1-p}}. \end{aligned}$$

We take now $\tau =\frac{r'}{2}$ and $d(x)\le \frac{r}{4}$ and we derive by the mean value theorem

$$\begin{aligned} u(x)\le c'_4r'^{-\frac{1}{q+1-p}}d(x)=c'_4d(x)(d_K(x))^{-\frac{1}{q+1-p}}, \end{aligned}$$

(2.36)

with $c'_4=c'_4(p,q)>0$ Letting $r'\rightarrow \frac{7}{8}r$, we get (2.12).

Step 2: global estimates If $d(x)\ge \frac{1}{4}d_K(x)$, there holds

$$\begin{aligned} d(x)(d_K(x))^{-\frac{1}{q+1-p}}\ge 2^{-\frac{2}{q+1-p}}(d(x))^{\frac{q-p}{q+1-p}}. \end{aligned}$$

Combining this inequality with (2.6) and obtain (2.34). $\square$

Remark

Under the assumption of Proposition 2.8, it follows from the maximum principle that u is upper bounded in the set $\Omega '_{r^*}:=\{x\in \Omega :d(x)>r^*\}=\Omega {\setminus }\overline{\Omega }_{r^*}$ by the solution w of

$$\begin{aligned} -\Delta _pw+|\nabla w|^q= & {} 0\quad \text {in }\Omega _{r^*}\nonumber \\ w= & {} c_4d(x)(d_K(x))^{-\frac{1}{q+1-p}}\quad \text {in }\partial \Omega _{r^*}, \end{aligned}$$

(2.37)

and w itself is bounded by $d^*=\max \{cd(x)(d_K(x))^{-\frac{1}{q+1-p}}:d(x)=r^*\}$.

Next we prove a boundary Harnack inequality. We recall that $\delta _1$ has been introduced at Corollary 2.3, and that the interior and exterior sphere conditions hold in the set $\{x\in \mathbb {R}^N:d (x)\le \delta _1\}$.

Theorem 2.9

Let $q>p-1$ and $0\in \partial \Omega$. Then there exists $c_5=c_5(N,p,q,\Omega )>0$ such that for any positive solution $u\in C(\Omega \cup ((\partial \Omega {\setminus }\{0\})\cap B_{2\delta _1})\cap C^1(\Omega )$ of (1.1) in $\Omega$, vanishing on $\partial \Omega {\setminus }\{0\})\cap B_{2\delta _1}$, there holds

$$\begin{aligned} {\frac{u(y)}{c_5d(y)} }\le {\frac{u(x)}{d(x)} }\le c_5{\frac{u(y)}{d(y)} } \end{aligned}$$

(2.38)

for all $x,y\in B_{\frac{2\delta _1}{3}}\cap \Omega$ such that $\frac{1}{2}\left| x\right| \le \left| y\right| \le 2\left| x\right|$.

For proving Theorem 2.9 we need some intermediate lemmas. First we recall the following result from [1].

Lemma 2.10

Assume that $a \in \partial \Omega$, $0<r<\delta _1$ and $h>1$ is an integer. There exists an integer $N_0$, depending only on $\delta _1$, such that for any points x and y in $\Omega \cap B_{\frac{3r}{2}}(a)$ verifying $\min \{d(x),d(y)\} \ge r/2^h$, there exists a connected chain of balls $B_1,\ldots ,B_j$ with $j\le N_0h$ such that

$$\begin{aligned} x \in B_1, y \in B_j, \quad B_i\cap B_{i+1} \ne \emptyset \quad \text {for } 1\le i \le j-1 \nonumber \\ \quad \text {and}\quad 2B_i \subset B_{2r}(Q) \cap \Omega \quad \text {for } 1\le i \le j. \end{aligned}$$

(2.39)

The next result is a standard Harnack inequality.

Lemma 2.11

Assume $a \in (\partial \Omega \setminus \{0\}) \cap B_{\frac{2\delta _1}{3}}$ and $0<r\le \left| a\right| /4$. Let $u\in C(\Omega \cup ((\partial \Omega \setminus \{0\})\cap B_{2\delta _1}))\cap C^1(\Omega )$ be a positive solution of (1.1) vanishing on $(\partial \Omega \setminus \{0\})\cap B_{2\delta _1}$. Then there exists a positive constant $c_6>1$ depending on N,  p,  q and $\delta _1$ such that

$$\begin{aligned} u(x) \le c_6^h u(y), \end{aligned}$$

(2.40)

for every $x,y \in B_{\frac{3r}{2}}(a)\cap \Omega$ such that $\min \{d(x),d(y)\} \ge r/2^h$ for some $h \in \mathbb N$.

Proof

For $\ell >0$, we define $T_{\ell }[u]$ by

$$\begin{aligned} T_\ell [u](x)=\ell ^{\frac{p-q}{q+1-p}}u(\ell x), \end{aligned}$$

(2.41)

and we notice that if u satisfies (1.1) in $\Omega$, then $T_\ell [u]$ satisfies the same equation in $\Omega ^{\ell }:=\ell ^{-1}\Omega$. If we take in particular $\ell =|a|$, we can assume $|a|=1$, thus the curvature of the domain $\Omega ^{|a|}$ remains bounded. By Proposition 2.8

$$\begin{aligned} u(x) \le c'_6\quad \forall x \in B_{2r}(a)\cap \Omega \end{aligned}$$

(2.42)

where $c'_6$ depends on N, q, $\delta _1$. Then we proceed as in [11], using Lemma 2.10 and internal Harnack inequality as quoted in [16, Corollary 10]. $\square$

Since the solutions are Hölder continuous, the following statement holds as in [16, Theorem 4.2]:

Lemma 2.12

Let the assumptions on a and u of Lemma 2.11 be fulfilled. If $b \in \partial \Omega \cap B_r(a)$ and $0<s\le 2^{-1}r$, there exist two positive constants $\delta$ and $c_7$ depending on N, p, q and $\Omega$ such that

$$\begin{aligned} u(x) \le c_7{\frac{\left| x-b\right| ^\delta }{s^\delta } }\max \{u(z): z\in B_r(b)\cap \Omega \} \end{aligned}$$

(2.43)

for every $x \in B_s(b) \cap \Omega$.

As a consequence we derive the following Carleson type estimate.

Lemma 2.13

Assume $a \in (\partial \Omega \setminus \{0\}) \cap B_{\frac{2\delta _1}{3}}$ and $0<r\le \left| a\right| /8$. Let $u\in C(\Omega \cup ((\partial \Omega \setminus \{0\})\cap B_{2\delta _1}))\cap C^2(\Omega )$ be a positive solution of (1.1) vanishing on $(\partial \Omega \setminus \{0\})\cap B_{2\delta _1}$. Then there exists a constant $c_8$ depending only on N, p and q such that

$$\begin{aligned} u(x) \le c_8 u\left( a-\frac{r}{2}\mathbf{n}_{_a}\right) \quad \forall x \in B_r(a) \cap \Omega . \end{aligned}$$

(2.44)

Proof

By Lemma 2.11 it is clear that for any integer h and $x \in B_r(a) \cap \Omega$ such that $d(x)\ge 2^{-h}r$, there holds

$$\begin{aligned} u(x) \le c_6^h u\left( a-\frac{r}{2}\mathbf{n}_{_a}\right) . \end{aligned}$$

(2.45)

Therefore u satisfies inequality (2.43) as any Hölder continuous function does. The proof that the constant is independent of r and u is more delicate. It is done in [1, Lemma 2.4] for linear equations, but it is based only on Lemma 2.12 and a geometric construction, thus it is also valid in our case. $\square$

Lemma 2.14

Assume $a \in (\partial \Omega \setminus \{0\}) \cap B_{\frac{2\delta _1}{3}}$ and $0<r\le \left| a\right| /8$. Let $u\in C(\Omega \cup ((\partial \Omega \setminus \{0\})\cap B_{2\delta _1}))\cap C^2(\Omega )$ be a positive solution of (1.1) vanishing on $(\partial \Omega \setminus \{0\})\cap B_{2\delta _1}$. Then there exist $\alpha \in (0,1/2)$ and $c_9>0$ depending on N, p and q such that

$$\begin{aligned} {\frac{1}{c_9} }{\frac{t}{r} } \le {\frac{u(b-t\mathbf{n}_{_b})}{u(a-\frac{r}{2}\mathbf{n}_{_a})} } \le c_9{\frac{t}{r} } \end{aligned}$$

(2.46)

for any $b \in B_r(a) \cap \partial \Omega$ and $0 \le t < \frac{\alpha }{2}r$.

Proof

It is similar to the one of [11, Lemma 3.15]. $\square$

Proof of Theorem 2.9

Assume $x \in B_{\frac{2\delta _1}{3}} \cap \Omega$ and set $r=\frac{\left| x\right| }{8}$.

Step 1: tangential estimate: we suppose $d(x) < \frac{\alpha }{2}r$. Let $a \in \partial \Omega \setminus \{0\}$ such that $\left| a\right| =\left| x\right|$ and $x \in B_r(a)$. By Lemma 2.14,

$$\begin{aligned} {\frac{8}{c_9} }{\frac{u(a-\frac{r}{2}\mathbf{n}_{_a})}{\left| x\right| } } \le {\frac{u(x)}{d(x)} } \le 8c_9{\frac{u(a-\frac{r}{2}\mathbf{n}_{_a})}{\left| x\right| } }. \end{aligned}$$

(2.47)

We can connect $a-\frac{r}{2}\mathbf{n}_{_a}$ with $-2r\mathbf{n}_{_0}$ by $m_1$ (depending only on N) connected balls $B_i=B_\frac{r}{4}(x_i)$ with $x_i \in \Omega$ and $d(x_i) \ge \frac{r}{2}$ for every $1 \le i \le m_1$. It follows from (2.44) that

$$\begin{aligned} c^{-m_1}_6u(-2r\mathbf{n}_{_0}) \le u\left( a-\frac{r}{2}\mathbf{n}_{_a}\right) \le c^{m_1}_6u(-2r\mathbf{n}_{_0}), \end{aligned}$$

which, together with (2.47) leads to

$$\begin{aligned} {\frac{1}{c_{10}} }{\frac{u(-2r\mathbf{n}_{_0})}{\left| x\right| } } \le {\frac{u(x)}{d(x)} } \le c_{10}{\frac{u(-2r\mathbf{n}_{_0})}{\left| x\right| } }, \end{aligned}$$

(2.48)

with $c_{10}=8c_9c^{m_1}_6$.

Step 2: internal estimate: we suppose $d(x) \ge \frac{\alpha }{2}r$. We can connect $-2r\mathbf{n}_{_0}$ with x by $m_2$ (depending only on N) connected balls $B'_i=B_\frac{\alpha r}{4}(x'_i)$ with $x'_i \in \Omega$ and $d(x'_i) \ge \frac{\alpha }{2}r$ for every $1 \le i \le m_2$. By Harnack and Carleson inequalities (2.40) and (2.44) and since $\frac{\alpha }{4}\left| x\right| <d(x)\le \left| x\right|$, we get

$$\begin{aligned} {\frac{\alpha }{4c_{6}'^{m_2}} }{\frac{u(-2r\mathbf{n}_{_0})}{\left| x\right| } } \le {\frac{u(x)}{d(x)} } \le {\frac{4c_{6}'^{m_2}}{\alpha } }{\frac{u(-2r\mathbf{n}_{_0})}{\left| x\right| } }. \end{aligned}$$

(2.49)

Step 3: end of proof Suppose $\frac{\left| x\right| }{2}\le s \le 2\left| x\right|$, we can connect $-2r\mathbf{n}_{_Q}$ with $-s\mathbf{n}_{_Q}$ by $m_3$ (depending only on N) connected balls $B''_i=B_\frac{r}{2}(x''_i)$ with $x''_i \in \Omega$ and $d(x''_i) \ge r$ for every $1 \le i \le m_3$. This fact, jointly with (2.48) and (2.49), yields to

$$\begin{aligned} {\frac{1}{c_{11}} }{\frac{u(-s\mathbf{n}_{_0})}{\left| x\right| } } \le {\frac{u(x)}{d(x)} } \le c_{11}{\frac{u(-s\mathbf{n}_{_0})}{\left| x\right| } } \end{aligned}$$

(2.50)

where $c_{11}=c_{11}(N,q,\Omega )$. Finally, if $y \in B_{\frac{2r_0}{3}} \cap \Omega$ satisfies $\frac{\left| x\right| }{2} \le \left| y\right| \le 2\left| x\right|$, then by applying twice (2.50) we get (2.38) with $c_{5}=c_{11}^2$. $\square$

The following inequality is a consequence of Theorem 2.9.

Corollary 2.15

Assume $q>p-1$ and $0\in \partial \Omega$. Then there exists $c_{12}>0$ depending on p, q and $\Omega$ such that for any positive solutions $u_1,\,u_2\in C(\Omega \cup ((\partial \Omega {\setminus }\{0\})\cap B_{2\delta _1}))\cap C^1(\Omega )$ of (1.1) in $\Omega$, vanishing on $(\partial \Omega {\setminus }\{0\})\cap B_{2\delta _1}$, there holds

$$\begin{aligned} \sup \left\{ {\frac{u_1(y)}{u_2(y)} }:y\in B_r{\setminus } B_{\frac{r}{2}}\right\} \le c_{12}\inf \left\{ {\frac{u_1(y)}{u_2(y)} }:y\in B_r{\setminus } B_{\frac{r}{2}}\right\} . \end{aligned}$$

(2.51)

3 Boundary singularities

3.1 Strongly singular solutions

In this section we consider the Eq. (1.1) in $\mathbb {R}_+^N$. We denote by $(r,\sigma ) \in \mathbb {R}_+ \times S^{N-1}$ the spherical coordinates in $\mathbb {R}^N$ and

$$\begin{aligned} S_+^{N-1}=\left\{ (\sin \phi \sigma ',\cos \phi ):\sigma '\in S^{N-2},\phi \in [0,\frac{\pi }{2})\right\} . \end{aligned}$$

If $v(x)=r^{-\beta }\omega (\sigma )$ satisfies (1.1) in $\mathbb {R}_+^N$ and vanishes on $\partial {\mathbb {R}^N_+}{\setminus }\{0\}$, then $\beta =\beta _q$ and $\omega$ is a solution of

$$\begin{aligned}&-div'\left( \left( \beta _q^2\omega ^2+|\nabla ' \omega |^2\right) ^{\frac{p-2}{2}}\nabla ' \omega \right) -\beta _q\Lambda _{\beta _q}\left( \beta _q^2\omega ^2+|\nabla ' \omega |^2\right) ^{\frac{p-2}{2}}\omega \nonumber \\&\quad +\left( \beta _q^2\omega ^2+|\nabla ' \omega |^2\right) ^{\frac{q}{2}}=0\quad \text{ in } S_+^{N-1}\\&\quad \omega =0\quad \text{ on } \partial S_+^{N-1}\nonumber \end{aligned}$$

(3.1)

where $\beta _q$ and $\Lambda _{\beta _q}$ have been defined in (1.10). We denote by $(\beta _*,\psi _*)\in \mathbb {R}_+^*\times C^2(\overline{S}_+^{N-1})$ the unique couple such $\max \psi _*=1$ with the property that the function $(r,\sigma )\mapsto r^{-\beta _*}\psi _*(\sigma )$ is positive, p-harmonic in $\mathbb {R}_+^N$ and vanishes on $\partial {\mathbb {R}^N_+}{\setminus }\{0\}$. Then $\psi _*=\psi$ satisfies

$$\begin{aligned}&\displaystyle -div'\left( \left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\nabla ' \psi \right) -\beta _*\Lambda _{\beta _*}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi =0\quad \text{ in } S_+^{N-1} \nonumber \\&\displaystyle \psi =0\quad \text{ on } \partial S_+^{N-1}. \end{aligned}$$

(3.2)

Since the function $\psi _*$ is unique it depends only on the azimuthal variable $\theta _{N-1}=\cos ^{-1}(\frac{x_N}{|x|})$ (see Appendix B). Our first result is the following

Theorem 3.1

If $q\ge q_*$, or equivalently $\beta _q\le \beta _*$, there exists no positive solution to problem (3.1).

Proof

Suppose such a solution $\omega$ exists and put $\theta =\beta _q/\beta _*$, then $0<\theta \le 1$. Set $\eta =\psi ^\theta$, where $\psi$ is a positive solution of (3.2), and define the operator ${\mathcal T}$ by

$$\begin{aligned} {\mathcal T}(\eta )&=-div'\left( \left( \beta _q^2\eta ^2+|\nabla ' \eta |^2\right) ^{\frac{p-2}{2}}\nabla ' \eta \right) -\beta _q\Lambda _{\beta _q}\left( \beta _q^2\eta ^2+|\nabla ' \eta |^2\right) ^{\frac{p-2}{2}}\eta \nonumber \\&\quad +\left( \beta _q^2\eta ^2+|\nabla ' \eta |^2\right) ^{\frac{q}{2}}. \end{aligned}$$

(3.3)

Since $\nabla \eta =\theta \psi ^{\theta -1}\nabla \psi$,

$$\begin{aligned} \left( \beta _q^2\eta ^2+|\nabla ' \eta |^2\right) ^{\frac{p-2}{2}}&=\theta ^{p-2}\psi ^{(\theta -1)(p-2)}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}},\\ \left( \beta _q^2\eta ^2+|\nabla ' \eta |^2\right) ^{\frac{p-2}{2}}\nabla ' \eta&=\theta ^{p-1}\psi ^{(\theta -1)(p-1)}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\nabla '\psi , \end{aligned}$$

therefore

$$\begin{aligned} {\mathcal T}(\eta )&=-\theta ^{p-1}\psi ^{(\theta -1)(p-1)}\mathrm{div'}\left( \left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\nabla '\psi \right) \\&\quad - \theta ^{p-1}(\theta -1)(p-1)\psi ^{(\theta -1)(p-1)-1}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}|\nabla '\psi |^2\\&\quad -\beta _q\Lambda _{\beta _q}\theta ^{p-2}\psi ^{(\theta -1)(p-1)}\left( \beta _*^2\psi ^2{+}|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi {+}\theta ^{q}\psi ^{(\theta -1)q}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{q}{2}}. \end{aligned}$$

But $\beta _q\Lambda _{\beta _q}\theta ^{p-2}=\beta _*\Lambda _{\beta _q}\theta ^{p-1}\le \beta _*\Lambda _{\beta _*}\theta ^{p-1}$ since $\beta _q\le \beta _*$. Using (3.2), we see that ${\mathcal T}(\eta )\ge 0$. Because Hopf Lemma is valid, there holds $\partial _\mathbf{n}\psi <0$ on $\partial S^{N-1}_+$. Since $\omega$ is $C^1$ in $\overline{S^{N-1}_+}$ and $\psi$ is defined up to an homothety, there exists a smallest function $\psi$ such that $\eta \ge \omega$, and the graphs of $\eta$ and $\omega$ over $\overline{S^{N-1}_+}$ are tangent, either at some $\alpha \in S^{N-1}_+$, or only at a point $\alpha \in \partial S^{N-1}_+$. We put $w=\eta -\omega$. Then

$$\begin{aligned} {\mathcal T}(\eta )={\mathcal T}(\eta )-{\mathcal T}(\omega )=\Phi (1)-\Phi (0), \end{aligned}$$

(3.4)

where $\Phi (t)={\mathcal T}(\omega _t)$ with $\omega _t=\omega +tw$.

We use local coordinates $(\sigma _1,\ldots ,\sigma _{N-1})$ on $S^{N-1}$ near $\alpha$. We denote by $g=(g_{ij})$ the metric tensor on $S^{N-1}$ and by $g^{jk}$ its contravariant components. Then, for any $\varphi \in C^1(S^{N-1})$,

$$\begin{aligned} {\left| \nabla \varphi \right| ^{2}}= \sum _{j,k}g^{jk}\frac{\partial \varphi }{\partial \sigma _{j}}\frac{\partial \varphi }{\partial \sigma _{k}}=\langle \nabla \varphi ,\nabla \varphi \rangle _g. \end{aligned}$$

If $X=(X^1,\ldots , X^d)\in C^1(TS^{N-1})$ is a vector field, we lower indices by setting ${X^\ell =\sum _{i}g^{\ell i}X_{i}}$ and define the divergence of X by

$$\begin{aligned} div'_gX=\frac{1}{\sqrt{\left| g\right| }} \sum _{\ell } \frac{\partial }{\partial \sigma _{\ell }}\left( \sqrt{\left| g\right| }X^\ell \right) =\frac{1}{\sqrt{\left| g\right| }} \sum _{\ell ,i}\frac{\partial }{\partial \sigma _{\ell }}\left( \sqrt{\left| g\right| }g^{\ell i}X_{i}\right) . \end{aligned}$$

We write $\Phi (t)=\Phi _1(t)+\Phi _2(t)+\Phi _3(t)$ where

$$\begin{aligned} \Phi _1(t)=-\beta _q\Lambda _{\beta _q}\left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-2}{2}}\omega _t,\quad \Phi _2(t)=\left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{q}{2}} \end{aligned}$$

and

$$\begin{aligned} \Phi _3(t)=-\mathrm {div'}\left( \left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-2}{2}}\nabla ' \omega _t\right) . \end{aligned}$$

Then

$$\begin{aligned} \Phi _1(1)-\Phi _1(0)=-\sum _{j}a_{j}\frac{\partial w}{\partial \sigma _{j}}-bw\quad \text {and}\quad \Phi _2(1)-\Phi _2(0)=\sum _{j}c_{j}\frac{\partial w}{\partial \sigma _{j}}+dw, \end{aligned}$$

where

$$\begin{aligned} b&=\beta _q\Lambda _{\beta _q}\left( \beta _q^{2}{\omega _t}^2+\left| \nabla \omega _t\right| ^2\right) ^{\frac{p}{2}-2} \left( (p-1)\beta _q^2\omega _t^2+\left| \nabla \omega _t\right| ^2\right) ,\\ a_{j}&=(p-2)\beta _q\Lambda _{\beta _q}\left( \beta _q^{2}{\omega _t}^2+\left| \nabla \omega _t\right| ^2\right) ^{\frac{p}{2}-2}\omega _t\sum _{k}g^{jk} {\frac{\partial \omega _t}{\partial \sigma _k} },\\ d&=q\beta _q^{2}\left( \beta ^{2}{\omega _t}^2+\left| \nabla \omega _t\right| ^2\right) ^{\frac{q}{2}-1}\omega _t, \end{aligned}$$

and

$$\begin{aligned} c_{j}=q\left( \beta _q^{2}{\omega _t}^2+\left| \nabla \omega _t\right| ^2\right) ^{\frac{q}{2}-1}\sum _{k}g^{jk} {\frac{\partial \omega _t}{\partial \sigma _k} }. \end{aligned}$$

Furthermore

$$\begin{aligned} \Phi _3(1)-\Phi _3(0)&=-(p-2)\mathrm{div'}\left( \left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-4}{2}}\left( \beta _q^2\omega _t w+ \langle \nabla '\omega _t,\nabla ' w\rangle _g\right) \nabla '\omega _t\right) \\&\quad -\mathrm{div'}\left( \left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-2}{2}}\nabla ' w\right) . \end{aligned}$$

Therefore we can write $\Phi (1)-\Phi (0)$ under the form

$$\begin{aligned} \Phi (1)-\Phi (0)=-\mathrm{div'}(A\nabla 'w)+\langle B,\nabla ' w\rangle _g+Cw:={\mathcal L}w \end{aligned}$$

(3.5)

where

$$\begin{aligned} \langle AX,X\rangle _g&=\left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-4}{2}} \left( p-2)\langle \nabla '\omega _t,X\rangle _g^2+|\nabla ' \omega _t|^2|X|^2\right) \nonumber \\&\ge \left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-4}{2}}\min \{1,p-1\}|\nabla ' \omega _t|^2|X|^2. \end{aligned}$$

(3.6)

and B and C can be computed from the previous expressions. It is important to notice that $\beta _q^2\omega _t^2+|\nabla ' \omega _t|^2$ is bounded between two positive constants $m_1$ and $m_2$ in $\overline{S^{N-1}_+}$. Thus the operator ${\mathcal L}$ is uniformly elliptic with bounded coefficients. Since w is nonnegative and either at some point $\alpha$, $\nabla 'w(\alpha )=0$ and $w(\alpha )>0$, or at some boundary point $\alpha$ where $w(\alpha )=0$ and $\partial _\mathbf{n}w(\alpha )<0$, it follows from the strong maximum principle or Hopf Lemma (see [7]) that $w=0$, contradiction. $\square$

Theorem 3.2

Assume $q<q_*$ or equivalently $\beta _q> \beta _*$. There exists a unique positive solution $\omega _*$ to problem (3.1).

Proof

Existence It will follow from [4]. Indeed problem (3.1) can be written under the form

$$\begin{aligned} \begin{array}{lll} \mathbf{A}(\omega ):=-div'\,\mathbf{a}(\omega ,\nabla '\omega )&{}=\mathbf{B}(\omega ,\nabla '\omega )&{}\quad \text {in }S^{N-1}_+\\ \omega &{}=0&{}\quad \text {on }\partial S^{N-1}_+, \end{array} \end{aligned}$$

(3.7)

where

$$\begin{aligned} \mathbf{a}(r,\xi )&=\left( \beta _q^2r^2+|\xi |^2\right) ^{\frac{p-2}{2}}\xi ,\nonumber \\ \mathbf{B}(r,\xi )&=\beta _q\Lambda _{\beta _q}\left( \beta _q^2r^2+|\xi |^2\right) ^{\frac{p-2}{2}}r -\left( \beta _q^2r^2+|\xi |^2\right) ^{\frac{q}{2}}. \end{aligned}$$

(3.8)

The operator $\mathbf{A}$ is a Leray–Lions operator which satisfies the assumptions (1.6)–(1.8) of [4, Theorem 2.1], and the term $\mathbf{B}$ satisfies (1.9), (1.10) in the same article. Therefore the existence of a positive solution $\omega \in W^{1,p}_0(S^{N-1}_+)\cap L^\infty (S^{N-1}_+)$ is ensured whenever we can find a supersolution $\overline{\omega }\in W^{1,p}(S^{N-1}_+)\cap L^\infty (S^{N-1}_+)$ and a nontrivial subsolution $\underline{\omega }\in W^{1,p}(S^{N-1}_+)$ of (3.7) such that

$$\begin{aligned} 0\le \underline{\omega }\le \overline{\omega }\quad&\text {in }S^{N-1}_+. \end{aligned}$$

(3.9)

First we note that $\eta =\eta _0$ is a supersolution if the positive constant $\eta _0$ is large enough. In order to find a subsolution, we set again $\eta =\psi ^\theta$ with $\theta =\beta _q/\beta _*$ and $\psi$ as in (3.2). Now $\theta > 1$, thus $\eta \in W^{1,p}_0(S^{N-1}_+)$. As above we have

$$\begin{aligned} {\mathcal T}(\eta )&=-\theta ^{p-1}\psi ^{(\theta -1)(p-1)}\mathrm{div'}\left( \left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\nabla '\psi \right) \\&\quad -\theta ^{p-1}(\theta -1)(p-1)\psi ^{(\theta -1)(p-1)-1}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}|\nabla '\psi |^2\\&\quad -\beta _q\Lambda _{\beta _q}\theta ^{p-2}\psi ^{(\theta -1)(p-1)}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi {+}\theta ^{q}\psi ^{(\theta -1)q}\left( \beta _*^2\psi ^2{+}|\nabla ' \psi |^2\right) ^{\frac{q}{2}}. \end{aligned}$$

Now $\beta _q\Lambda _{\beta _q}\theta ^{p-2}=\beta _*\Lambda _{\beta _q}\theta ^{p-1}=\beta _*(\Lambda _{\beta _q}-\Lambda _{\beta _*})\theta ^{p-1}+\beta _*\Lambda _{\beta _*}\theta ^{p-1}$ and $\Lambda _{\beta _q}-\Lambda _{\beta _*}=(\beta _q-\beta _*)(p-1)=\beta _*(p-1)(\theta -1)$, hence

$$\begin{aligned} {\mathcal T}(\eta )&=- \theta ^{p-1}\psi ^{(\theta -1)(p-1)}\mathrm{div'}\left( \left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\nabla '\psi \right) \\&\quad -\theta ^{p-1}(\theta -1)(p-1)\psi ^{(\theta -1)(p-1)-1}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}|\nabla '\psi |^2\\&\quad -\beta _*(\Lambda _{\beta _q}-\Lambda _{\beta _*})\theta ^{p-1}\psi ^{(\theta -1)(p-1)}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi \\&\quad -\beta _*\Lambda _{\beta _*}\theta ^{p-1}\psi ^{(\theta -1)(p-1)}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi {+}\theta ^{q}\psi ^{(\theta -1)q}\left( \beta _*^2\psi ^2{+}|\nabla ' \psi |^2\right) ^{\frac{q}{2}}. \end{aligned}$$

Using the equation satisfied by $\psi$ yields to the relation

$$\begin{aligned} {\mathcal T}(\eta )&= -\theta ^{p-1}(\theta -1)(p-1)\psi ^{(\theta -1)(p-1)-1}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}|\nabla '\psi |^2\\&\quad -\beta _*^2(p-1)(\theta -1)\theta ^{p-1}\psi ^{(\theta -1)(p-1)-1}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-2}{2}}\psi ^2\\&\quad +\theta ^{q}\psi ^{(\theta -1)q}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{q}{2}}\\&=-\theta ^{p-1}(\theta -1)(p-1)\psi ^{(\theta -1)(p-1)-1}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p}{2}}\\&\quad +\theta ^{q}\psi ^{(\theta -1)q}\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{q}{2}}. \end{aligned}$$

If we replace $\eta :=\eta _1=\psi ^\theta$ by $\eta :=\eta _m=(m\psi )^\theta$ in the above computation, the inequality ${\mathcal T}\eta _m)\le 0$ will be true provided

$$\begin{aligned} m^{\theta (q+1-p)}\psi ^{(\theta -1)(q+1-p)+1}\le \theta ^{p-1-q}(\theta -1)(p-1)\left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-q}{2}}, \end{aligned}$$

which is satisfied if we choose m small enough so that $(m\psi )^\theta \le \eta _0$ and satisfying

$$\begin{aligned} m^{\theta (q+1-p)}\le \beta _*^{(\theta -1)(q+1-p)+1}\theta ^{p-1-q}(\theta -1)(p-1)\frac{\min _{x\in S_+^{N-1}} \left( \beta _*^2\psi ^2+|\nabla ' \psi |^2\right) ^{\frac{p-q}{2}}}{\max _{x\in S_+^{N-1}} \psi ^{(\theta -1)(q+1-p)+1}}. \end{aligned}$$

Therefore $0<\eta _m\le \eta _0$ and standard regularity implies that the solution $\omega$ is $C^{1}$ in $\overline{S}^{N-1}_+$. Actually $\omega$ is $C^{\infty }$ since the operator is not degenerate.

Uniqueness We use the tangency method developed in the proof of Theorem 3.1. Assume $\omega _1$ and $\omega _2$ are two positive solutions of (3.2), then they are positive in $S^{N-1}_+$ and $\partial _\mathbf{n}\omega _i<0$ on $\partial S^{N-1}_+$. Either the $\omega _i$ are ordered and $\omega _1\le \omega _2$, or their graphs intersect. In any case we can define

$$\begin{aligned} \tau =\inf \{s>1:s\omega _1\ge \omega _2\}. \end{aligned}$$

We set $\omega ^*=\tau \omega _1$. Then either the graphs of $\omega _2$ and $\omega ^*$ are tangent at some interior point $\alpha$, or they are not tangent in $S^{N-1}_+$, $\partial _\mathbf{n}\omega ^*\le \partial _\mathbf{n}\omega _2<0$ on $\partial S^{N-1}_+$ and there exists $\alpha \in \partial S^{N-1}_+$ such that $\partial _\mathbf{n}\omega ^*(\alpha )= \partial _\mathbf{n}\omega _2(\alpha )<0$. Furthermore ${\mathcal T}(\omega ^*)\ge 0$. If we set $w=\omega ^*-\omega _2$, then, as in Theorem 3.1,

$$\begin{aligned} -\mathrm{div'}(\tilde{A}\nabla 'w)+\langle \tilde{B},\nabla ' w\rangle _g+\tilde{C}w=\tilde{\mathcal L}w\ge 0 \end{aligned}$$

where

$$\begin{aligned} \langle \tilde{A}X,X\rangle _g&=\left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-4}{2}} \left( p-2)\langle \nabla '\omega _t,X\rangle _g^2+|\nabla ' \omega _t|^2|X|^2\right) \nonumber \\&\ge \left( \beta _q^2\omega _t^2+|\nabla ' \omega _t|^2\right) ^{\frac{p-4}{2}}\min \{1,p-1\}|\nabla ' \omega _t|^2|X|^2, \end{aligned}$$

(3.10)

in which $\omega _t=\omega _2+t(\omega ^*-\omega _2)$ and $t\in (0,1)$ is obtained by applying the mean value theorem and $\tilde{B}$ and $\tilde{C}$ are defined accordingly. Since $\tilde{\mathcal L}$ is uniformly elliptic and has bounded coefficients, it follows from the strong maximum principle that $w=0$. Thus $\omega ^*=\tau \omega _1=\omega _2$ and $\tau =1$ from the equation. This ends the proof. $\square$

3.2 Removable boundary singularities

The following is the basic result for removability of isolated singularities. It is valid in the general case, but with a local geometric constraint.

Theorem 3.3

Assume $q^*\le q<p\le N$, $\Omega$ is a $C^2$ bounded domain with $0\in \partial \Omega$, such that $\Omega \cap B_\delta =B_\delta ^+$ for some $\delta >0$. If $u\in C^1(\overline{\Omega }{\setminus }\{0\})$ is a nonnegative solution of (1.1) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$, then it is identically 0.

Proof

Step 1: assume $\Omega \subset \mathbb {R}^N_+$ For $\epsilon >0$, we set $\Omega '_\epsilon =\Omega \cap \overline{B^c_\epsilon }$ and $H_\epsilon =\mathbb {R}^N_+\cap \overline{B^c_\epsilon }$. For $k,n\in \mathbb N_*$, $n\ge \mathrm{diam\,}(\Omega )$, we denote by $v_{k,n,\epsilon }$ ($n\in \mathbb N_*$) the solution of the problem

$$\begin{aligned} -\Delta _pv+\left| \nabla v\right| ^q&=0\qquad \qquad \;\;\;\text {in }H_\epsilon \cap B_n\nonumber \\ v&=k\chi _{_{\mathbb {R}^N_+\cap \partial B_\epsilon }}\quad \text {on }\partial (H_\epsilon \cap B_n). \end{aligned}$$

(3.11)

If $k>c_2\epsilon ^{\frac{q-p}{q+1-p}}$ for a suitable $c_2=c_2(p,q)>0$ (see Lemma 2.6), then $v_{k,n,\epsilon }\ge u$ in $\Omega '_\epsilon$. Moreover there holds $v_{k,n,\epsilon }\le v_{k',n',\epsilon }$ for $n\le n'$ and $k\le k'$. Furthermore the function

$$\begin{aligned} U_{\epsilon ,n}(x)=c_2\left( (\left| x\right| -\epsilon )^{\frac{q-p}{q+1-p}}-(n-\epsilon )^{\frac{q-p}{q+1-p}}\right) \end{aligned}$$

is a supersolution in $B_n{\setminus } B_\epsilon$, and there holds $v_{k,n,\epsilon }\le U_{\epsilon ,n}$. By monotonicity and standard a priori estimate, we obtain that $v_{k,n,\epsilon }\rightarrow v_\epsilon$ when $n, k\rightarrow \infty$ and that the function $v=v_\epsilon$ is solution of

$$\begin{aligned} -\Delta _pv+\left| \nabla v\right| ^q&=0\qquad \text {in }H_\epsilon \nonumber \\ \lim \nolimits _{\left| x\right| \rightarrow \epsilon }v(x)&=\infty \nonumber \\ v&=0\qquad \text {on }\partial \mathbb {R}^N_+\cap \overline{B^c_\epsilon }. \end{aligned}$$

(3.12)

Furthermore

$$\begin{aligned} u(x)\le v_\epsilon (x)\le c_2(\left| x\right| -\epsilon )^{\frac{q-p}{q+1-p}}\quad \text {in }\Omega '_\epsilon . \end{aligned}$$

(3.13)

The function $v_\epsilon$ may not be unique, however it is the minimal solution of the above problem since the $v_{k,n,\epsilon }$ is unique, and monotonicity in n and k holds. Actually, $v_\epsilon \le v_{\epsilon '}$ if $0\le \epsilon \le \epsilon '$. For $\ell >0$, we recall that the transformation $v\mapsto T_\ell [v]$ defined by (2.41) leaves Eq. (1.1) invariant. As a consequence of the uniqueness of the approximations we have $T_\ell [v_{k,n,\epsilon }]=v_{ \ell ^{\frac{p-q}{q+1-p}}k,\ell ^{-1}n,\ell ^{-1}\epsilon }$, which implies

$$\begin{aligned} T_\ell [v_\epsilon ]=v_{\ell ^{-1}\epsilon }. \end{aligned}$$

(3.14)

Letting $\epsilon \rightarrow 0$, we derive from the monotonicity with respect to $\epsilon$ and standard $C^{1,\alpha }$ estimates, that the following identity holds:

$$\begin{aligned} T_\ell [v_0]=v_0\quad \forall \ell >0. \end{aligned}$$

(3.15)

The function $v_0$ is a positive and separable solution of (1.1) in $\mathbb {R}^N_+$ which vanishes on $\partial \Omega {\setminus }\{0\}$. It follows from Theorem 3.1 that $v_0=0$, and so is u.

Step 2: the general case We assume that $\Omega \cap B_\delta \subset \mathbb {R}^N_+$ and we denote by M the maximum of u on $\partial B_\delta \cap \Omega$. Then the function $(u-M)_+$ is a subsolution of (1.1) in $\Omega \cap B_\delta$ which vanishes on $\partial \Omega \cap B_\delta {\setminus }\{0\}$. By Step 1, it is dominated by $v_0$, which ends the proof. $\square$

Remark

The previous result is valid if u is a subsolution with the same regularity. If u is no longer assumed to be nonnegative, only $u^+$ vanishes. Furthermore, the regularity of the boundary has not been used, but only the fact that $\Omega$ is locally contained into a half space to the boundary of which 0 belongs.

Remark

If no geometric assumption is made on $\partial \Omega$, we can prove that $u(x)=o (\left| x\right| ^{-\beta _q})$ near 0. The next result shows that the removability holds if $q>q_*$.

Theorem 3.4

Assume $q^*< q<p\le N$ and $\Omega$ is a $C^2$ bounded domain with $0\in \partial \Omega$. If u is a nonnegative solution of (1.1) in $\Omega$ which belongs to $C^1(\overline{\Omega }{\setminus }\{0\})$ and vanishes on $\partial \Omega {\setminus }\{0\}$, it is identically 0.

Proof

As it is proved in [12], for any smooth subdomain $S\subset S^{N-1}$, there exists a unique $\beta _{*\,s}>0$ and $\psi ^*_s>0$, unique up to an homothety, such that $x\mapsto \left| x\right| ^{-\beta _{*\,s}}\psi ^*_s(\left| x\right| ^{-1}x)$ is p harmonic in the cone $C_S=\{x\in \mathbb {R}^N{\setminus }\{0\}:\left| x\right| ^{-1}x\in S\}$ and $\psi ^*_s$ satisfies

$$\begin{aligned}&\displaystyle -div'\left( \left( \beta _{*\,s}^2\psi ^{*\,2}_s+|\nabla ' \psi ^*_s|^2\right) ^{\frac{p-2}{2}}\nabla ' \psi ^*_s\right) -\beta _{*\,s}\Lambda _{\beta _{*\,s}}\left( \beta _{*\,s}^2\psi ^{*\,2}+|\nabla ' \psi ^*_s|^2\right) ^{\frac{p-2}{2}}\psi ^*_s=0\quad \text{ in } S\nonumber \\&\displaystyle \psi ^*_s=0\quad \text{ on } \partial S, \end{aligned}$$

(3.16)

Furthermore $S\subset \tilde{S}\subset S^{N-1}$ implies $\beta _{*\,\tilde{s}}\le \beta _{*\,s}$. Using the system of spherical coordinates defined in (6.5) in Appendix B, for $\epsilon >0$ we denote by $S:=S_\epsilon$ the spherical shell with vertex the north pole N and latitude angle $\theta _{N-1}\in [0,\frac{\pi }{2}+\epsilon ]$. Because of uniqueness of $\beta _{*\,s}$, $\beta _{*\,s_\epsilon }\uparrow \beta _*$ as $\epsilon \rightarrow 0$. Therefore, if $q>q_*$, or equivalently $\beta _q<\beta _*$, there exists $\delta ,\epsilon >0$ such that $\Omega \cap B_\delta \subset C_{S_\epsilon }\cap B_\delta$ and $\beta _q<\beta _{*\,s_\epsilon }$. Since 3.1 is valid if $S^{N-1}_+$ is replaced by $S_\epsilon$ and $\beta _q<\beta _{*\,s_\epsilon }$ it follows that $u=0$ as in the proof of Theorem 3.3, Steps 1 and 2. $\square$

The next result, valid in the case $p=N$, is based upon the conformal invariance of the N-Laplacian. In this case the exponent $\beta _*$ corresponding to the first spherical N-harmonic eigenvalue is equal to 1 and the corresponding spherical N-harmonic eigenfunction in $S^{N-1}_+$ is $x_N/\left| x\right| ^{2}$.

Theorem 3.5

Assume $N-\frac{1}{2}\le q< N$, $\Omega$ is a bounded domain and $0\in \partial \Omega$ is such that there exists a ball $B\subset \Omega ^c$ to the boundary of which 0 belongs. If u is a nonnegative solution of

$$\begin{aligned} -\Delta _Nu+\left| \nabla u\right| ^q=0\quad \text {in }\Omega , \end{aligned}$$

(3.17)

which belongs to $C(\overline{\Omega }{\setminus }\{0\})\cap W^{1,N}_0(\Omega {\setminus } \overline{B}_\epsilon (0))$ for any $\epsilon >0$, it is identically 0.

Proof

We assume that the inward normal unit vector to $\partial \Omega$ at 0 is $\mathbf{e}_N=(0,0,\ldots ,1)$ and that the ball $B=B_{\frac{1}{2}}(a)$ of center $a=-\frac{1}{2}\mathbf{e}_N$ and radius $\frac{1}{2}$ touches $\partial \Omega$ at 0 and is exterior to $\Omega$ (this can be assumed up to a rotation and a dilation). This is the consequence of the exterior sphere condition at the point 0. It is always valid if $\partial \Omega$ is $C^2$. We denote by ${\mathcal I}_\omega$ the inversion of center $\omega =-\mathbf{e}_N$ and power 1, i.e. ${\mathcal I}_\omega (x)=\omega +\frac{x-\omega }{\left| x-\omega \right| ^2}$. Under this transformation, the complement of the ball $B_{\frac{1}{2}}(a)$, which contains $\Omega$, is transformed into the half space $\mathbb {R}^N_-$ which contains the image $\tilde{\Omega }$ of $\Omega$. Since u satisfies (3.17), $\tilde{u}=u\circ {\mathcal I}_\omega$ satisfies

$$\begin{aligned} -\Delta _N\tilde{u}+\left| x-\omega \right| ^{2(q-N)}\left| \nabla \tilde{u}\right| ^q=0\quad \text {in }\tilde{\Omega }. \end{aligned}$$

(3.18)

Furthermore since $0={\mathcal I}_\omega (0)$ and ${\mathcal I}_\omega$ is a diffeomorphism, $\tilde{u}\in C(\overline{\tilde{\Omega }}{\setminus }\{0\})\cap C^1(\tilde{\Omega })$ and it vanishes on $\partial \tilde{\Omega }{\setminus }\{0\}$. Since $\left| x-\omega \right| \le 1$ and $q<N$, $\tilde{u}$ is a subsolution for (3.17) in $\tilde{G}$. By Theorem 3.4, $\tilde{u}=0$. $\square$

3.3 Weakly singular solutions

The main result of this section is the following existence and uniqueness result concerning solutions of (1.1) with a boundary weak singularity. We recall that $\psi _*$ is unique positive solution of (1.11) such that $\sup \psi _*=1$. Our first result is valid for any $1<p\le N$ but it needs a geometric constraint on $\Omega$.

Theorem 3.6

Let $p-1<q<q_*<p\le N$ and $\Omega \subset \mathbb {R}^N_+$ be a bounded $C^2$ domain such that $0\in \partial \Omega$. Assume that there exists $\delta >0$ such that $\Omega _\delta :=\Omega \cap B_\delta = B^+_\delta$. Then for any $k>0$ there exists a unique positive solution $u:=u_k$ of (1.1) in $\Omega$, which belongs to $C^1(\overline{\Omega }{\setminus }\{0\})$, vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{u_k(x)}{\Psi _*(x)}=k \end{aligned}$$

(3.19)

in the $C^1$-topology of $S^{N-1}_+$, where

$$\begin{aligned} \Psi _*(x)=\left| x\right| ^{-\beta _*}\psi _*(|x|^{-1}x). \end{aligned}$$

The proof of this theorem is long and difficult and requires a certain number of intermediate results.

Lemma 3.7

Let the assumptions on p, q and $\Omega$ of Theorem 3.6 be satisfied. There exists a unique positive p-harmonic function $\Phi _{*}$ in $\Omega$, which is continuous in $\overline{\Omega }{\setminus }\{0\}$, vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{\Phi _{*}(x)}{\Psi _*(x)}=1. \end{aligned}$$

(3.20)

Proof

For $0<\epsilon <\delta$ let $v_\epsilon$ be the unique nonnegative p-harmonic function in $\Omega {\setminus } \overline{B^+_{\epsilon }}$ which is continuous in $\overline{\Omega }{\setminus } {B^+_{\epsilon }}$, vanishes on $\partial \Omega {\setminus } {B_{\epsilon }}$ and achieves the value ${\Psi _*}$ on $\partial B_\epsilon \cap \Omega$. Since $\Omega \subset \mathbb {R}^N_+$, $v_\epsilon \le \Psi _*$ in $\Omega {\setminus } {B^+_{\epsilon }}$. Hence inequalities $0<\epsilon <\epsilon '\le \delta$ imply $v_\epsilon \le v_{\epsilon '}$ in $\Omega {\setminus } \overline{B^+_{\epsilon '}}$. Because $\Psi _*\le \delta ^{-\beta _*}$, there holds

$$\begin{aligned} v_\epsilon +\delta ^{-\beta _*}\ge \Psi _*, \end{aligned}$$

(3.21)

in $\Omega {\setminus } B_\delta ^+$. Since $v_\epsilon$ and $\Psi _*$ coincide on $\partial B_\epsilon ^+$ and vanish on $\partial \mathbb {R}^N_+\cap (B_\delta ^+ {\setminus } B_\epsilon ^+)$, (3.21) holds also in $B_\delta ^+ {\setminus } B_\epsilon ^+$. Because $v_\epsilon \ge 0$ there holds

$$\begin{aligned} (\Psi _*-\delta ^{-\beta _*})_+\le v_\epsilon \le \Psi _*\quad \text{ in } \Omega {\setminus } B_\epsilon ^+. \end{aligned}$$

(3.22)

By a standard regularity result $v_\epsilon$ converges to a function $\Phi _{*}$ continuous in $\overline{\Omega }{\setminus }\{0\}$, p-harmonic in $\Omega$ such that

$$\begin{aligned} (\Psi _*-\delta ^{-\beta _*})_+\le \Phi _{*}\le \Psi _* \end{aligned}$$

in $\Omega$. Therefore (3.20) holds provided $\frac{x}{\left| x\right| }$ remains in a compact subset of $S^{N-1}_+$. Let us define a function $\tilde{\phi }_{*}$ by $\tilde{\phi }_*(x)=\left| x\right| ^{\beta _*}\Phi _*(x)$, then $\tilde{\phi }_*(r,\sigma )\le {\psi _*}(\sigma )$ where $r=\left| x\right|$ and $\sigma =\frac{x}{\left| x\right| }\in S^{N-1}_+$. By standard $C^{1,\alpha }$ estimates, $\tilde{\phi }_*(r,.)$ is relatively compact in the $C(\overline{S^{N-1}_+})$-topology. Therefore the convergence of $\frac{\Phi _*(x)}{{\Psi _*}(x)}$ to 1 when x to 0 holds not only when $\frac{x}{\left| x\right| }$ remains in a compact subset of $S^{N-1}_+$, but uniformly on $S^{N-1}_+$, which implies (3.20). Uniqueness follows classically by (3.20) and the maximum principle. $\square$

Lemma 3.8

Let the assumptions on p, q and $\Omega$ of Theorem 3.6 be satisfied. If for some $k>0$ there exists a solution $u_{k}$ of (1.1) in $\Omega$, which belongs to $C^1(\overline{\Omega }{\setminus }\{0\})$, vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies (3.19), then for any $k>0$ there exists such a solution.

Proof

We notice that for any $c<1$ (resp $c>1$), $cu_{k}$ is a subsolution (resp. supersolution) of (1.1) in $\Omega$. Let $\Phi _*$ be as in Lemma 3.7. If $c<1$, the function $ck\Phi _*$ is a supersolution of (1.1) which vanishes on $\partial \Omega {\setminus }\{0\}$. Furthermore

$$\begin{aligned} \lim _{x\rightarrow 0}{\frac{cu_{k}(x)}{\Psi _*(x)} }=ck=\lim _{x\rightarrow 0}{\frac{ck\Phi _*(x)}{\Psi _*(x)} }. \end{aligned}$$

Then there exists a solution $u_{ck}$ of (1.1) in $\Omega$ which satisfies $cu_{k}\le u_{ck}\le ck\Phi _*$. If $c>1$, we set $u^*:=T_{c^\theta }[u_k]$, which means $u^*(x)= c^{\beta _q\theta }u_k(c^\theta \,x)$ with $\theta =(\beta _q-\beta _*)^{-1}$. Then $u^*$ is a solution of (1.1) in $\Omega ^{c^\theta }=\frac{1}{c^\theta }\Omega$. In particular, $u^*$ satisfies the equation in $B^+_{\frac{\delta }{c^\theta }}(0)$. Since $c^\theta >1$, $B^+_{\frac{\delta }{c^\theta }}(0)\subset B^+_{\delta }(0)$. Put $m=\max \{u^*:x\in \partial B^+_{\frac{\delta }{c^\theta }}(0)\}$. The function $(u^*-m)_+$, extended by 0 outside $B^+_{\frac{\delta }{c^\theta }}(0)$, is a subsolution of (1.1) in $\Omega$. Furthermore it satisfies

$$\begin{aligned} \lim _{x\rightarrow 0}{\frac{(u^*-m)_+(x)}{\Psi _*(x)} }=ck, \end{aligned}$$

uniformly on any compact subset of $S^{N-1}_+$. Therefore there exists a solution $u_{ck}$ of (1.1) in $\Omega$ which satisfies $(u^*-m)_+\le u_{ck}\le ck\Phi _*$, and in particular it vanishes on $\partial \Omega {\setminus }\{0\}$ and belongs to $C^1(\overline{\Omega }{\setminus }\{0\})$. By [13], $u_{ck}$ is positive in $\Omega$. Thus $u_{ck}$ belongs to $C^{1,\alpha }(\overline{B^+_\delta }(0){\setminus }\{0\})$ and satisfies

$$\begin{aligned} \left| x\right| ^{\beta _*}\left| u_{ck}(x)\right| +\left| x\right| ^{1+\beta _*}\left| \nabla u_{ck}(x)\right| +\left| x\right| ^{1+\beta _*+\alpha }\sup _{\begin{array}{c} |y|\le |x|\\ x\ne y \end{array}} \frac{\left| \nabla u_{ck}(x)-\nabla u_{ck}(y)\right| }{|x-y|^\alpha }\le M \end{aligned}$$

by (2.11). Therefore the set of functions $\{r^{\beta _*+1}\nabla u_{ck}(r,\cdot )\}_{r>0}$ is uniformly relatively compact in the topology of uniform convergence on $\overline{S}_+^{N-1}$. Since it converges to $ck\nabla '\psi _*$ uniformly on compact subsets of $S_+^{N-1}$ as $r\rightarrow 0$, this convergence holds in $C(\overline{S_+^{N-1}})$. This implies

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{u_{ck}(x)}{\Psi _*(x)}=ck. \end{aligned}$$

(3.23)

$\square$

The next Lemma is the keystone of our construction. Its proof is very delicate and needs several intermediate steps.

Lemma 3.9

Under the assumptions of Theorem 3.6 there exists a real number $R_0$ such that $0<R_0\le \delta$ and a positive subsolution $\tilde{u}$ of (1.1) in $B^+_{R_0}$ which is Lipschitz continuous in $\overline{B^+_{R_0}}{\setminus }\{0\}$, vanishes on $\overline{B^+_{R_0}}\cap \partial \mathbb {R}^N_+{\setminus }\{0\}$, is smaller than $\Psi _*$ and satisfies

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{\tilde{u}(x)}{\Psi _*(x)}=1. \end{aligned}$$

(3.24)

Proof

The construction of the function $\tilde{u}$. We look for a subsolution under the form $\tilde{u}=\Psi _*-w$ for a suitable nonnegative function w.

Step 1: reduction of the problem We use spherical coordinates for a $C^1$ function $u:x\mapsto u(x)=u(r,\sigma )$, $r=|x|$, $\sigma =\frac{x}{|x|}$. Then $\nabla u=u_r\mathbf{e}+r^{-1}\nabla ' u$ where $\mathbf{e}=\left| x\right| ^{-1}x$, $\left| \nabla u\right| ^2= u^2_r+r^{-2}\left| \nabla ' u\right| ^2$ and $\left| \nabla u\right| ^q=\left( u^2_r+r^{-2}\left| \nabla ' u\right| ^2\right) ^{\frac{q}{2}}$. The expression of the p-Laplacian in spherical coordinates is

$$\begin{aligned} -\Delta _p u&=-\left( \left( u^2_r+r^{-2}\left| \nabla ' u\right| ^2\right) ^{\frac{p-2}{2}}u_r\right) _r -{\frac{N-1}{r} }\left( u^2_r+r^{-2}\left| \nabla ' u\right| ^2\right) ^{\frac{p-2}{2}}u_r\\&\quad -{\frac{1}{r^2} } div'\left( \left( u^2_r+r^{-2}\left| \nabla ' u\right| ^2\right) ^{\frac{p-2}{2}}\nabla ' u\right) . \end{aligned}$$

Put $v(t,\sigma )=r^{\beta _*}u(r,\sigma )$ with $t=\ln r\in (-\infty ,\ln \delta ]$, then v satisfies

$$\begin{aligned}&{\mathcal Q}[v]:=\nonumber \\&\quad -\left( \left( (v_t-\beta _*v)^2+\left| \nabla ' v\right| ^2\right) ^{\frac{p-2}{2}}(v_t-\beta _*v)\right) _t\nonumber \\&\qquad -div'\left( \left( (v_t-\beta _*v)^2+\left| \nabla ' v\right| ^2\right) ^{\frac{p-2}{2}}\nabla 'v\right) \nonumber \\&\qquad +\Lambda _{\beta _*}\left( (v_t-\beta _*v)^2+\left| \nabla ' v\right| ^2\right) ^{\frac{p-2}{2}}(v_t-\beta _*v)+e^{\nu t}\left( (v_t-\beta _*v)^2+\left| \nabla ' v\right| ^2\right) ^{\frac{q}{2}}=0 \end{aligned}$$

(3.25)

in $(-\infty ,\ln \delta )\times S^{N-1}_+$ where $\nu =1-(q+1-p)(\beta _*+1)=1-\frac{\beta _*+1}{\beta _q+1}>0$ and $\Lambda _{\beta _*}=\beta _*(p-1)+p-N$. Notice that $\psi _*$ satisfies

$$\begin{aligned} -div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla ' \psi _*\right| ^2\right) ^{\frac{p-2}{2}}\nabla '\psi _*\right) -\beta _*\Lambda _{\beta _*}\left( \beta _*^2\psi _*^2+\left| \nabla ' \psi _*\right| ^2\right) ^{\frac{p-2}{2}}\psi _*=0,\nonumber \\ \end{aligned}$$

(3.26)

hence it is a supersolution for (3.25). We look for a subsolution under the form

$$\begin{aligned} V(t,\sigma )=\psi _*-a(t)g(\psi _*) \end{aligned}$$

where g is a continuous increasing function defined on $\mathbb {R}_+$, vanishing at 0 and smooth on $\mathbb {R}^*_+$ and $a(t)=e^{\gamma t}$ with $\gamma >0$ to be chosen. Thus $a'=\gamma a$, $a''=\gamma ^2 a$, $V_t=-\gamma ag(\psi _*)$, $V_t-\beta _*V=-\beta _*\psi _*+a(\beta _*-\gamma )g(\psi _*)$, $\nabla ' V=(1-ag'(\psi _*))\nabla '\psi _*$ and

$$\begin{aligned}&(V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2=\left( -\beta _*\psi _*+a(\beta _*-\gamma )g(\psi _*)\right) ^2+(1-ag'(\psi _*))^2\left| \nabla '\psi _*\right| ^2\\&\quad =\left( \beta _*^2\psi _*^2+2a\beta _*(\gamma -\beta _*)g(\psi _*)\psi _*\right) +\left( 1-2ag'(\psi _*)\right) \left| \nabla '\psi _*\right| ^2 +O(a^2\left\| g(\psi )\right\| _{C^1})\\&\quad =\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2+2a\left( \beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2\right) +O(a^2\left\| g(\psi _*)\right\| _{C^1}). \end{aligned}$$

Therefore

$$\begin{aligned}&\left( (V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2\right) ^{\frac{p-2}{2}}\\&\quad =\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\left[ 1+(p-2)a{\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\right] \\&\qquad +O(a^2\left\| g(\psi )\right\| _{C^1}), \end{aligned}$$

and

$$\begin{aligned}&e^{\nu t}\left( (V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2\right) ^{\frac{q}{2}} \\&\quad =e^{\nu t}\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{q}{2}}\left[ 1+qa{\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\right] \\&\qquad +O(e^{\nu t}a^2\left\| g(\psi _*)\right\| _{C^1}), \end{aligned}$$

thus

$$\begin{aligned}&\left( (V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2\right) ^{\frac{p-2}{2}}(V_t-\beta _*V)\\&\quad =-\beta _*\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\psi _*+ a(\beta _* -\gamma )\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g(\psi _*)\\&\qquad -a\beta _*(p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{4-p}{2}}} } \psi _* +O(a^2\left\| g(\psi _*)\right\| _{C^1}). \end{aligned}$$

Finally,

$$\begin{aligned}&-\left( \left( (V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2\right) ^{\frac{p-2}{2}}(V_t-\beta _*V)\right) _t\nonumber \\&\quad =a\left[ (\gamma ^2-\beta _* \gamma )\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g(\psi _*)\right. \nonumber \\&\qquad \left. +\beta _*(p-2){\frac{\beta _*(\gamma ^2-\beta _*\gamma )\psi _*g(\psi _*)-\gamma g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{4-p}{2}}} } \psi _*\right] +O(a^2\left\| g(\psi _*)\right\| _{C^2}). \end{aligned}$$

(3.27)

Since

$$\begin{aligned}&\left( (V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2\right) ^{\frac{p-2}{2}}\nabla 'V\\&\quad =\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}(1-ag'(\psi _*))\\&\qquad \times \left[ 1+a(p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\right] \nabla '\psi _*\\&\qquad +O(a^2\left\| g(\psi _*)\right\| _{C^1})\\&\quad =\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\nabla '\psi _*\\&\qquad +a\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\\&\qquad \times \left[ (p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} } -g'(\psi _*)\right] \nabla '\psi _*\\&\qquad +O(a^2\left\| g(\psi _*)\right\| _{C^1}), \end{aligned}$$

we get similarly

$$\begin{aligned}&-div'\left( \left( (V_t-\beta _*V)^2+\left| \nabla ' V\right| ^2\right) ^{\frac{p-2}{2}}\nabla 'V\right) = -div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\nabla '\psi _*\right) \nonumber \\&-a\,div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\left[ (p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} } -g'(\psi _*)\right] \nabla '\psi _*\right) \nonumber \\&\quad +O(a^2\left\| g(\psi _*)\right\| _{C^2}). \end{aligned}$$

(3.28)

Noting that

$$\begin{aligned} -div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\nabla '\psi _*\right) \psi _* =\beta _*\Lambda _{\beta _*}\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\psi _*,\nonumber \\ \end{aligned}$$

(3.29)

we obtain

$$\begin{aligned}&e^{-\gamma t}{\mathcal Q}[V]\nonumber \\&\quad =\left[ (\gamma ^2-\beta _* \gamma )\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g(\psi _*)\right. \nonumber \\&\qquad \left. +\, \beta _*(p-2){\frac{\beta _*(\gamma ^2-\beta _*\gamma )\psi _*g(\psi _*)-\gamma g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{(\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2)^{\frac{4-p}{2}}} } \psi _*\right] \nonumber \\&\qquad -div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\right. \nonumber \\&\qquad \times \left. \left[ (p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} } -g'(\psi _*)\right] \nabla '\psi _*\right) \nonumber \\&\qquad -\Lambda _{\beta _*}\left( (\gamma -\beta _* )\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g(\psi _*)\right. \nonumber \\&\qquad \left. +\, \beta _*(p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{(\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2)^{\frac{4-p}{2}}} }\psi _*\right) \nonumber \\&\qquad +e^{(\nu -\gamma ) t}\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{q}{2}}\left[ 1+qa{\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\right] \nonumber \\&\qquad +O(a\left\| g(\psi _*)\right\| _{C^2}). \end{aligned}$$

(3.30)

In this expression we have in particular

$$\begin{aligned}&-div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\right. \nonumber \\&\qquad \left. \times \left[ (p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} } -g'(\psi _*)\right] \nabla '\psi _*\right) \nonumber \\&\quad =(p-1)div'\left[ g'(\psi _*)\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\nabla \psi _*\right] \nonumber \\&\qquad -\beta _*div'\left( \left( \beta _*^2\psi _*^2\!+\!\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-4}{2}}\left[ (p-2)\beta _*\psi _*g'(\psi _*)\!+\!(p-2)(\gamma -\beta _*)g(\psi _*)\right] \psi _*\right) \nonumber \\&\quad =(p-1)g''(\psi _*)\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}|\nabla \psi _*|^2\nonumber \\&\qquad +(p-1)g'(\psi _*)div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\nabla \psi _*\right) \nonumber \\&\qquad -(p-2)\beta _*div'\left[ {\frac{\left( (\gamma -\beta _*) g(\psi _*)\psi _*+\beta _*g'(\psi _*)\psi ^2_*\right) }{\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{4-p}{2}}} } \nabla '\psi _*\right] . \end{aligned}$$

(3.31)

Using the Eq. (3.26) satisfied by $\psi _*$, it infers that

$$\begin{aligned}&-div'\left( \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\right. \nonumber \\&\qquad \times \left. \left[ (p-2){\frac{\beta _*(\gamma -\beta _*)\psi _*g(\psi _*)-g'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} } -g'(\psi _*)\right] \nabla '\psi _*\right) \nonumber \\&\quad =(p-1)\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}\left( g''(\psi _*)|\nabla '\psi _*|^2 -\beta _*\Lambda _{\beta _*}g'(\psi _*)\psi _*\right) \nonumber \\&\quad -(p-2)\beta _*div'\left[ {\frac{\left( (\gamma -\beta _*) g(\psi _*)\psi _*+\beta _*g'(\psi _*)\psi ^2_*\right) }{\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{4-p}{2}}} } \nabla '\psi _*\right] . \end{aligned}$$

(3.32)

Plugging this identity into the expression (3.30), we obtain after some simplifications

$$\begin{aligned} e^{-\gamma t}{\mathcal Q}[V]=\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g(\psi _*){\mathcal Q}_1[V] +e^{(\nu -\gamma )t}R[V]+O(a\left\| g(\psi _*)\right\| _{C^2}), \end{aligned}$$

(3.33)

where

$$\begin{aligned} R[V]=e^{\nu t}\left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{q}{2}}\left[ 1+q{\frac{\beta _*(a'-\beta _*a)\psi _*g(\psi _*)-ag'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\right] , \end{aligned}$$

(3.34)

and

$$\begin{aligned} {\mathcal Q}_1[V]&= (\gamma -\Lambda _{\beta _*})(\gamma -\beta _*)\left[ 1+(p-2){\frac{\beta ^2_*\psi ^2_*}{\beta ^2_*\psi ^2_*+|\nabla '\psi _*|^2} }\right] -(p-1)\beta _*\Lambda _{\beta _*}{\frac{\psi _*g'(\psi _*)}{g(\psi _*)} }\nonumber \\&\quad +\left[ (p-4)\beta _*\Lambda _{\beta _*}\psi _*-2\Delta '\psi _*\right] \left( \gamma -\beta _*\left( 1-{\frac{\psi _*g'(\psi _*)}{g(\psi _*)} }\right) \right) {\frac{\beta _*\psi _*}{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\nonumber \\&\quad -(p-2)\left[ {\frac{\psi _*g'(\psi _*)}{g(\psi _*)} }\left( (\beta _*+1)\gamma -\beta _*\Lambda _{\beta _*}+\beta _*\right) +\gamma -\beta _*+\beta _*{\frac{\psi _*^2g''(\psi _*)}{g(\psi _*)} }\right] \nonumber \\&\quad \times {\frac{\left| \nabla '\psi _*\right| ^2}{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }+(p-1){\frac{g''(\psi _*)}{g(\psi _*)} }\left| \nabla '\psi _*\right| ^2. \end{aligned}$$

(3.35)

In this expression the difficult term to deal with is $\left[ (p-4)\beta _*\Lambda _{\beta _*}\psi _*-2\Delta '\psi _*\right]$ since it has not a prescribed sign. However $\Delta '\psi _*=O(\psi _*)$ by (6.19) in Appendix B.

Step 2: the perturbation method and the computation with $g(\psi _*)=\psi _*$ With such a choice of function g

$$\begin{aligned} {\mathcal Q}_1[V]&=(\gamma -\Lambda _{\beta _*})(\gamma -\beta _*)\left[ 1+(p-2){\frac{\beta ^2_*\psi ^2_*}{\beta ^2_*\psi ^2_*+|\nabla '\psi _*|^2} }\right] -(p-1)\beta _*\Lambda _{\beta _*}\nonumber \\&\quad -(p-2)\left[ (\gamma -\Lambda _{\beta _*})\beta _*+2\gamma \right] {\frac{\left| \nabla '\psi _*\right| ^2}{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }+\gamma \ O(\psi _*^2). \end{aligned}$$

(3.36)

Equivalently

$$\begin{aligned} {\mathcal Q}_1[V]&=\left[ 1+(p-2){\frac{\beta ^2_*\psi ^2_*}{\beta ^2_*\psi ^2_*+|\nabla '\psi _*|^2} }\right] \left( \gamma ^2-(\Lambda _{\beta _*}+\beta _*)\gamma \right) \\&\quad -\gamma \left[ (p-2)(\beta _*+2){\frac{\left| \nabla '\psi _*\right| ^2}{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }+O(\psi ^2_*)\right] \end{aligned}$$

and finally

$$\begin{aligned} {\mathcal Q}_1[V]=\left[ 1+(p-2){\frac{\beta ^2_*\psi ^2_*}{\beta ^2_*\psi ^2_*+|\nabla '\psi _*|^2} }\right] \gamma [\gamma -(\Lambda _{\beta _*}+\beta _*+(p-2)(\beta _*+2))+O(\psi ^2_*)]. \end{aligned}$$

(3.37)

Using the fact that $\beta _*>\frac{N-1}{p-1}$ if $1<p<2$ and $1<\beta _*<\frac{N-1}{p-1}$ if $2<p<N$ (see Theorem 6.1 in Appendix B), we have

$$\begin{aligned} \Lambda _{\beta _*}+\beta _*+(p-2)(\beta _*+2)\ge \left\{ \begin{array}{ll} \Lambda _{\beta _*}+\beta _*(p-1)&{}\quad \text {if }p\ge 2\\ N+3(p-2)>N-3&{}\quad \text {if }1<p<2. \end{array}\right. \end{aligned}$$

(3.38)

When $N=2$, we have explicitly $\beta _*=\frac{1+2\sqrt{p^2-3p+3}}{3(p-1)}$ (see [9, Th 3.3]). Therefore for all $N\ge 2$ and $p>1$, there holds

$$\begin{aligned} \Lambda _{\beta _*}+\beta _*+(p-2)(\beta _*+2)>0. \end{aligned}$$

(3.39)

We fix $\epsilon _0>0$ such that, whenever $\psi _*\le \epsilon _0$, there holds

$$\begin{aligned} \Lambda _{\beta _*}+\beta _*+(p-2)(\beta _*+2)+O(\psi ^2_*)>{\frac{1}{2} }\left( \Lambda _{\beta _*}+\beta _*+(p-2)(\beta _*+2)\right) . \end{aligned}$$

(3.40)

If we fix $\gamma _0>0$ such that

$$\begin{aligned} \gamma _0<\min \left\{ {\frac{1}{2} }\left( \Lambda _{\beta _*}+\beta _*+(p-2)(\beta _*+2)\right) ,\nu ,\beta _*\right\} , \end{aligned}$$

(3.41)

we obtain

$$\begin{aligned} {\mathcal Q}_1[V]\le -\min \{1,p-1\}\gamma m^2\quad \forall \, 0<\gamma \le \gamma _0, \end{aligned}$$

(3.42)

whenever $\psi _*\le \epsilon _0$, for some m depending only on p, q and N (through $\psi _*$ and $\nu$), which, in the same range of value of $\psi _*$, yields to

$$\begin{aligned} \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g(\psi _*){\mathcal Q}_1[V]\le -c_{17} \psi _*\quad \forall \, 0<\gamma \le \gamma _0, \end{aligned}$$

(3.43)

for some $c_{17}>0$ depending on N, p, q. This estimate is valid whatever is $p>1$, but only in a neighborhood of $\psi _*=0$. If we replace $g(\psi _*)=\psi _*$ by $g_k(\psi _*)=\psi _*e^{-k\psi _*}$ for $0<k<1$, and denote by ${\mathcal Q}_{1,k}[V]$ the corresponding expression of ${\mathcal Q}_1[V]$ which becomes now ${\mathcal Q}_{1,0}[V]$. We define similarly ${\mathcal Q}_k[V]$, and ${\mathcal Q}[V]$ becomes ${\mathcal Q}_0[V]$. Since $g'_k(\psi _*)=e^{-k\psi _*}-kg_k(\psi _*)$ and $g''_k=-2ke^{-k\psi _*}+k^2g_k(\psi _*)$, we obtain

$$\begin{aligned} {\mathcal Q}_{1,k}[V]&={\mathcal Q}_{1,0}[V]+k(p-1)\beta _*\Lambda _{\beta _*}\psi _*+(p-1)\left( -{\frac{2k}{\psi _*} }+k^2\right) \left| \nabla '\psi _*\right| ^2\nonumber \\&\quad +(2-p)\beta _*\left( -2k+k^2\right) \psi _*+O(\psi _*^2) \end{aligned}$$

(3.44)

Notice that $\nabla '\psi _*$ vanishes only at the North pole $\mathbf{e}_N$, thus there exists $k_0\in (0,1]$ such that

$$\begin{aligned} k(1-p)\beta _*\Lambda _{\beta _*}\psi _*+(p-1)\left( {\frac{2k}{\psi _*} }-k^2\right) \left| \nabla '\psi _*\right| ^2\ge {\frac{1}{2} }(2-p)_+\beta _*\left( -2k+k^2\right) \psi _*\quad \forall k{\le } k_0 \end{aligned}$$

whenever $\psi _*\le \epsilon _0$ which yields to

$$\begin{aligned} \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{p-2}{2}}g_k(\psi _*){\mathcal Q}_{1,k}[V]\le -c_{18}k\quad \forall k\le k_0 \end{aligned}$$

(3.45)

for some $c_{13}=c_{13}(N,p,q,\epsilon _0)$. There exists $c_{14}=c_{14}(N,p,q)>0$ such that

$$\begin{aligned} \left( \beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2\right) ^{\frac{q}{2}}\left[ 1+qe^{\gamma t}{\frac{\beta _*(\gamma -\beta _*)\psi _*g_k(\psi _*)-g_k'(\psi _*)\left| \nabla \psi _*\right| ^2 }{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\right] \le c_{14}\nonumber \\ \end{aligned}$$

(3.46)

in $S^{N-1}_+\times (-\infty ,\ln \delta ]$. Moreover

$$\begin{aligned} O(a\left\| g(\psi _*)\right\| _{C^2})\le e^{\gamma t}\tilde{c}_k \end{aligned}$$

(3.47)

for some $\tilde{c}_k=\tilde{c}_k(N,p,q)>0$. We derive from (3.45)–(3.47)

$$\begin{aligned} \begin{array}{ll} e^{-\gamma t}{\mathcal Q}_k[V]\le -c_{13}k+c_{14}e^{(\nu -\gamma ) t}+e^{\gamma t}\tilde{c}_k\qquad \forall k\le k_0 \end{array} \end{aligned}$$

(3.48)

Thus there exists $T_k\le \ln \delta$ such that ${\mathcal Q}_k[V]\le 0$, for all $t\le T_k$ and provided $\psi _*\le \epsilon _0$. This local estimate will be used in the construction of the subsolution when $p\ge 2$.

Step 3: the case $1<p<2$ Since the function $\psi ^*$ depends only on the azimuthal angle $\theta \in (0;\frac{\pi }{2}]$ we will write $\psi _*(\sigma )=\psi _*(\theta )$ and $\nabla '\psi _*(\sigma )=\psi _{*\theta }(\theta )\mathbf{n}$ where $\mathbf{n}$ is the downward unit vector tangent to $S^{N-1}$ in the hyperplane going through $\sigma$ and the poles. From (6.8),

$$\begin{aligned} (p-4)\beta _*\Lambda _{\beta _*}\psi _*-2\Delta '\psi _*=(p-2)\left( \beta _*\Lambda _{\beta _*}\psi _*+2{\frac{\beta _*^2\psi _*+{\psi _{*}}_{\theta \theta }}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\right) , \end{aligned}$$

(3.49)

since $\psi ^{\,2}_{*\theta }=\left| \nabla '\psi _*\right| ^2$ and thus

$$\begin{aligned}&\left( (p-4)\beta _*\Lambda _{\beta _*}\psi _*-2\Delta '\psi _*\right) {\frac{\beta _*\gamma \psi _*}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\nonumber \\&\quad =(p-2)\gamma \left( \Lambda _{\beta _*}{\frac{\beta _*^2\psi _*^2}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }+2\beta _*{\frac{\beta _*^2\psi _*^2+{\psi _{*}}_ {\theta \theta }\psi _*}{(\beta _*^2\psi ^2_{*}+\psi ^{\,2}_{*\theta })^2} } \right) . \end{aligned}$$

(3.50)

From Theorem 6.1-Step 4 in Appendix B, we know that $\beta _*^2\psi _*+{\psi _{*}}_{\theta \theta }\ge 0$, thus the contribution of this term to ${\mathcal Q}_1[V]$ is nonpositive. We replace this expression in ${\mathcal Q}_1[V]$ with $g(\psi _*)=\psi _*$ and obtain

$$\begin{aligned} {\mathcal Q}_1[V]&=(\gamma -\Lambda _{\beta _*})(\gamma -\beta _*)\left( 1+(p-2){\frac{\beta _*^2\psi ^2_{*}}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\right) -\Lambda _{\beta _*}\beta _*(p-1)\nonumber \\&\quad +(p-2)\gamma \Lambda _{\beta _*}{\frac{\beta _*^2\psi ^2_{*}}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }-(p-2)\left( (\beta _*+2)\gamma -\Lambda _{\beta _*}\beta _*\right) {\frac{\psi ^2_{*\theta }}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\nonumber \\&\quad +2\beta _*(p-2){\frac{\beta _*^2\psi _*^2+{\psi _{*}}_{\theta \theta }\psi _*}{(\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta })^2} }\gamma \nonumber \\&\le \gamma \left( 1+(p-2){\frac{\beta _*^2\psi ^2_{*}}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\right) \left( \gamma -\Lambda _{\beta _*}-\beta _*\right) \nonumber \\&\quad -(p-2)\gamma {\frac{(\beta _*+2))\psi ^2_{*\theta }-\Lambda _{\beta _*}\beta _*^2\psi _*^2}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\nonumber \\&\le \gamma \left( 1+(p-2){\frac{\beta _*^2\psi ^2_{*}}{\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\right) \nonumber \\&\quad \times \left( \gamma -\left( \Lambda _{\beta _*}\!+\beta _*+(p-2){\frac{(\beta _*+2)\psi ^2_{*\theta }-\Lambda _{\beta _*}\beta _*^2\psi _*^2}{(p-1)\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\right) \!\right) . \end{aligned}$$

(3.51)

We can write

$$\begin{aligned}&\Lambda _{\beta _*}+\beta _*+(p-2){\frac{(\beta _*+2)\psi ^2_{*\theta }-\Lambda _{\beta _*}\beta _*^2\psi _*^2}{(p-1)\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\nonumber \\&\quad ={\frac{\left( \Lambda _{\beta _*}+(p-1)\beta _*\right) \beta _*^2\psi _*^2+\left( \Lambda _{\beta _*}+\beta _*(p-1)+2(p-2)\right) \psi ^2_{*\theta }}{(p-1)\beta _*^2\psi ^2_{*}+\psi ^2_{*\theta }} }\nonumber \\&\quad \ge c_{15}\left( \Lambda _{\beta _*}+\beta _*(p-1)+2(p-2)\right) \end{aligned}$$

(3.52)

for some positive constant $c_{15}$. This expression $\Lambda _{\beta _*}+\beta _*(p-1)+2(p-2)$ is always positive: obviously if $N\ge 3$ and by using the explicit expression of $\beta _*$ if $N=2$. Thus there exists $\gamma _0$ and $c_{16}>0$ such that ${\mathcal Q}_1[V]<-c_{16}$ for $0<\gamma \le \gamma _0$. The perturbation method of Step 2, is valid in the whole range of values of $\psi _*$ and we derive from (3.42)–(3.43) that (3.48) holds for all $k\le k_0$ and $t\le T_k$. Therefore ${\mathcal Q}_k[V]\le 0$.

Step 4: the case $p\ge 2$ For $c>0$ to be fixed and $\psi _*\ge \epsilon _0$, $\gamma \in (0,\gamma _0]$, we take $g(\psi _*)=c\psi _*^{1-\frac{\gamma }{\beta _*}}$. Then we derive from (3.35):

$$\begin{aligned} {\mathcal Q}_1[V]&= (\gamma -\Lambda _{\beta _*})(\gamma -\beta _*){\frac{(p-1)\beta ^2_*\psi ^2_*+|\nabla '\psi _*|^2}{\beta ^2_*\psi ^2_*+|\nabla '\psi _*|^2} } -(p-1)\beta _*\Lambda _{\beta _*}\left( 1-{\frac{\gamma }{\beta _*} }\right) \nonumber \\&\quad -(p-1){\frac{\gamma (\beta _*-\gamma )}{\beta ^2_*} }\psi _*^{-1-\frac{\gamma }{\beta _*}}\left| \nabla '\psi _*\right| ^2 -(p-2)(\beta _*-\gamma )(\gamma -\Lambda _{\beta _*})\nonumber \\&\quad \times {\frac{\left| \nabla '\psi _*\right| ^2}{\beta _*^2\psi _*^2+\left| \nabla '\psi _*\right| ^2} }\nonumber \\&=(1-p)\left[ \gamma (\beta _*-\gamma )+{\frac{\gamma (\beta _*-\gamma )}{\beta ^2_*} }\psi _*^{-1-\frac{\gamma }{\beta _*}}\left| \nabla '\psi _*\right| ^2\right] . \end{aligned}$$

(3.53)

For $k\le k_0$ we fix c such that $c\epsilon _0^{1-\frac{\gamma }{\beta _*}}=\epsilon _0e^{-k\epsilon _0}\Longleftrightarrow c=\epsilon _0^{\frac{\gamma }{\beta _*}}e^{-k\epsilon _0}$ and we define g by

$$\begin{aligned} g(\psi _*)=\min \left\{ \psi _*e^{-k\psi _*},\epsilon _0^{\frac{\gamma }{\beta _*}}e^{-k\epsilon _0}\psi _*^{1-\frac{\gamma }{\beta _*}}\right\} {=}\left\{ \begin{array}{ll} \psi _*e^{-k\psi _*}&{}\quad \text {if }0\le \psi _*\le \epsilon _0\\ \epsilon _0^{\frac{\gamma }{\beta _*}}e^{-k\epsilon _0}\psi _*^{1-\frac{\gamma }{\beta _*}} &{}\quad \text {if }\epsilon _0\le \psi _*\le 1, \end{array}\right. \qquad \end{aligned}$$

(3.54)

and we set $V(t,\sigma )=\psi ^*(\sigma )-a(t)g(\psi _*(\sigma ))$ with $(t,\sigma )\in (-\infty ,T_k]\times S^{N-1}_+$ and define $\tilde{u}(r,\sigma )=r^{-\beta _*}(\psi ^*(\sigma )-a(\ln r)g(\psi _*(\sigma )))$ accordingly for $(r,\sigma )\in (-\infty ,e^{T_k}]\times S^{N-1}_+$. Since $\psi _*$ is a decreasing function the coincidence set $\{\sigma \in S^{N-1}_+:\psi _*(\sigma )=\epsilon _0\}$ is a circular cone $\Sigma _{\theta _0}$ with vertex 0, axis $\mathbf{e}_N$ and angle $\theta _0$. We set $R_0=e^{T_k}$

$$\begin{aligned} \begin{array} {ll}\Gamma _1=\left\{ x=(r,\theta )\in B^+_{R_0}:\theta _0< \theta <\frac{\pi }{2}\right\} =\left\{ (r,\sigma )\in [0,R_0)\times S^{N-1}_+:0<\psi _*(\sigma )<\epsilon _0\right\} ,\\ \Gamma _2=\left\{ x=(r,\theta )\in B^+_{R_0}:0< \theta <\theta _0\right\} =\left\{ (r,\sigma )\in [0,R_0)\times S^{N-1}_+:\epsilon _0<\psi _*(\sigma )<1\right\} , \end{array} \end{aligned}$$

and define

$$\begin{aligned} \tilde{u}(r,\sigma )&=r^{-\beta _*}\left( \psi _*(\sigma )-r^\gamma g(\psi _*(\sigma ))\right) \\&=\left\{ \begin{array}{ll} u_1(r,\sigma )=r^{-\beta _*}(1-r^\gamma e^{-k\psi _*(\sigma )})\psi _*(\sigma )&{}\quad \text {if }(r,\theta )\in \Gamma _1\\ u_2(r,\sigma )=r^{-\beta _*}\left( 1-r^\gamma \epsilon _0^{\frac{\gamma }{\beta _*}}e^{-k\epsilon _0}(\psi _*(\sigma ))^{1-\frac{\gamma }{\beta _*}}\right) \psi _*(\sigma )&{}\quad \text {if }(r,\theta )\in \Gamma _2. \end{array}\right. \end{aligned}$$

The function $\tilde{u}$ is a subsolution separately on $\Gamma _1$ and $\Gamma _2$ and is Lipschitz continuous in $\overline{\Omega }{\setminus }\{0\}$. If we denote by $g_1$ and $g_2$ the restriction of g to $\Gamma _1$ and $\Gamma _2$ respectively, that is to $\Omega _1$ and $\Omega _2$, then $g'_1(\epsilon _0)>g'_2(\epsilon _0)>0$. Let $\zeta \in C^{1}_c(B^+_{R_0})$ which vanishes in neighborhoods of 0 and $\partial B^+_{R_0}$, $\zeta \ge 0$, then

$$\begin{aligned} {\int _{\Gamma _i}^{}}\left| \nabla \tilde{u}\right| ^{p-2}\nabla \tilde{u}.\nabla \zeta dx+{\int _{\Omega _i}^{}}\left| \nabla \tilde{u}\right| ^{q}\zeta dx \le {\int _{\Sigma _{\theta _0}}^{}}\left| \nabla u_i\right| ^{p-2}\partial _{\mathbf{n}_i} u_i\zeta dS, \end{aligned}$$

(3.55)

where $\mathbf{n}_i$ is the normal unit vector on $\Sigma _{\theta _0}$ outward from $\Gamma _i$. Actually, $\mathbf{n}_2=-\mathbf{n}_1=\mathbf{n}$ thus

$$\begin{aligned} \nabla \tilde{u}=\tilde{u}_r\mathbf{e}+r^{-\beta _*-1}(1-r^\gamma g'(\psi _*))\nabla '\psi _*=\tilde{u}_r\mathbf{e}+r^{-\beta _*-1}(1-r^\gamma g'(\psi _*))\psi _{*\theta }\,\mathbf{n}. \end{aligned}$$

and on $\Sigma _{\theta _0}$,

$$\begin{aligned} \nabla \tilde{u}=\left\{ \begin{array} {ll}\tilde{u}_r\mathbf{e}-r^{-\beta _*-1}(1-r^\gamma g_1'(\epsilon _0))\psi _{*\theta }\,\mathbf{n}&{}\quad \text { in }\Gamma _1\\ \tilde{u}_r\mathbf{e}+r^{-\beta _*-1}(1-r^\gamma g_2'(\epsilon _0))\psi _{*\theta }\,\mathbf{n}&{}\quad \text { in }\Gamma _2 \end{array}\right. \end{aligned}$$

Therefore

$$\begin{aligned}&\left| \nabla u_1\right| ^{p-2}\partial _{\mathbf{n}_1}u_1\nonumber \\&\quad =-r^{-\beta _*-1}(1-r^\gamma g_1'(\epsilon _0))\left( \tilde{u}^2_r+r^{-2\beta _*-2}(1-r^\gamma g_1'(\epsilon _0))^2\psi ^2_{*\theta }\right) ^{\frac{p-2}{2}}\psi _{*\theta }\quad \text { in }\Gamma _1 \end{aligned}$$

and

$$\begin{aligned}&\left| \nabla u_2\right| ^{p-2}\partial _{\mathbf{n}_2}u_2\nonumber \\&\quad =r^{-\beta _*-1}(1-r^\gamma g_2'(\epsilon _0))\left( \tilde{u}^2_r+r^{-2\beta _*-2}(1-r^\gamma g_2'(\epsilon _0))^2\psi ^2_{*\theta }\right) ^{\frac{p-2}{2}}\psi _{*\theta }\quad \text { in }\Gamma _2. \end{aligned}$$

By adding the two inequalities (3.55)

$$\begin{aligned} {\int _{\Omega }^{}}\left| \nabla \tilde{u}\right| ^{p-2}\nabla \tilde{u}.\nabla \zeta dx+{\int _{\Omega }^{}}\left| \nabla \tilde{u}\right| ^{q}\zeta dx \le {\int _{\Sigma _{\theta _0}}^{}}\left( \left| \nabla u_1\right| ^{p-2}\partial _{\mathbf{n_1}} u_1+\left| \nabla u_2\right| ^{p-2}\partial _{\mathbf{n_2}} u_2\right) \zeta dS. \end{aligned}$$

(3.56)

By monotonicity of the function $X\mapsto \left( \tilde{u}^2_r+X^2\right) ^{\frac{p}{2}}$ and since

$$\begin{aligned} r^{-\beta _*-1}(1-r^\gamma g_2'(\epsilon _0))\ge r^{-\beta _*-1}(1-r^\gamma g_1'(\epsilon _0))\ge 0, \end{aligned}$$

we derive

$$\begin{aligned}&r^{-\beta _*-1}(1-r^\gamma g_2'(\epsilon _0))\left( \tilde{u}^2_r+r^{-2\beta _*-2}(1-r^\gamma g_2'(\epsilon _0))^2\psi ^2_{*\theta }\right) ^{\frac{p-2}{2}}\\&\quad \ge r^{-\beta _*-1}(1-r^\gamma g_1'(\epsilon _0))\left( \tilde{u}^2_r+r^{-2\beta _*-2}(1-r^\gamma g_1'(\epsilon _0))^2\psi ^2_{*\theta }\right) ^{\frac{p-2}{2}} \end{aligned}$$

We derive that the right-hand side of (3.56) is nonpositive because $\psi _{*\theta }\le 0$, and therefore $\tilde{u}$ is a positive subsolution of (1.1) in $B^+_{R_0}$ dominated by $\Psi _*$ and satisfying (3.24). $\square$

.

Proof of Theorem 3.6

Let $M=\max \{\Psi _*(x):x\in \partial B^+_{R_0}\}$, then $M=R^{-\beta _*}_0$. The function $u^{*}$ defined by

$$\begin{aligned} u^{*}(x)=\left\{ \begin{array} {ll} (\tilde{u}(x)-M)_+&{}\quad \text {if } x\in B^+_{R_0}\\ 0&{}\quad \text {if } x\in \Omega {\setminus } B^+_{R_0}, \end{array}\right. \end{aligned}$$

is indeed a subsolution of (1.1) in whole $\Omega$ where it satisfies $u^{*}\le \Psi _*$ and it vanishes on $\partial \Omega {\setminus }\{0\}$. Since $\Phi _*$ is a positive p-harmonic function in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies (3.20), it is supersolution of (1.1) and therefore it dominates $u^{*}$. Therefore there exists a solution u of (1.1) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies $u^{*}\le u\le \Phi _*$. This implies that (3.19) holds with $k=1$ and we conclude with Lemma 3.8. This ends the proof of Lemma 3.9. $\square$

When $p=N$ the statement of Theorem 3.6 holds without the flatness assumption on $\partial \Omega$. The proof of the next theorem is an easy adaptation to the one of Theorem 3.6, provided Lemmas 3.7, 3.8 and 3.9 are modified accordingly.

Theorem 3.10

Assume $N-1<q<N-\frac{1}{2}$ and $\Omega$ be a bounded $C^2$ domain such that $0\in \partial \Omega$. Then for any $k>0$ there exists a unique positive solution $u:=u_k$ of (3.17) in $\Omega$, which belongs to $C^1(\overline{\Omega }{\setminus }\{0\})$, vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies uniformly with respect to $\sigma \in S^{N-1}_+$

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow 0\\ x/\left| x\right| \rightarrow \sigma \end{array}}\left| x\right| u_k(x)=k\psi _*(\sigma ). \end{aligned}$$

(3.57)

Since $p=N$, then $\beta _*=1$ and $\psi _*(\sigma )=\frac{x_N}{\left| x\right| }=\cos \theta _{N-1}$ with the identification of $\sigma$ and $\theta _{N-1}:=\theta$. In a more intrinsic manner (3.57) can be written under the form

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow 0\\ x\in \Omega \end{array}}|x|^2\frac{u_k(x)}{ d(x)}=k. \end{aligned}$$

(3.58)

We recall that if $\omega \in \mathbb {R}^N$ and ${\mathcal I}_\omega$ denotes the inversion of center $\omega$ and power 1, i.e. ${\mathcal I}_\omega (x)=\omega +\frac{x-\omega }{|x-\omega |^2}$, then $\tilde{u}=u\circ {\mathcal I}_\omega$ satisfies (3.18).

Lemma 3.11

Assume $\Omega$ be a bounded $C^2$ domain such that $0\in \partial \Omega$. Then there exists a unique N-harmonic function $\Phi _*$ in $\Omega$, which vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow 0\\ x/\left| x\right| \rightarrow \sigma \end{array}}\left| x\right| \Phi _*(x)=\psi _*(\sigma ), \end{aligned}$$

(3.59)

uniformly with respect to $\sigma \in S^{N-1}_+$.

Proof

Uniqueness is standard. Let $\omega =-\mathbf{e}_N\in \overline{\Omega }^c$, with the notations of the proof of Theorem 3.5, $\omega '=-\omega$, $a=-\frac{1}{2}\mathbf{e}_N$ and $a'=-a$. We can assume that the balls $B_{\frac{1}{2}}(a)$ and $B_{\frac{1}{2}}(a')$ are tangent to $\partial \Omega$ at 0 and respectively subset of $\Omega ^c$ and $\Omega$. The function $x\mapsto \Psi (x)=-\frac{x_N}{|x|^2}$ which is N-harmonic in $\mathbb {R}^N_-$ and vanishes on $\partial \mathbb {R}^{N}_-{\setminus }\{0\}$ is transformed by the inversion ${\mathcal I}_{\omega '}$ of center $\omega '$ and power 1 into the function $\Psi _{\omega '}=\Psi \circ {\mathcal I}_\omega$ which is positive and N-harmonic in $B_{\frac{1}{2}}(a')$ and vanishes on $\partial B_{\frac{1}{2}}(a'){\setminus }\{0\}$. The function $\hat{\Psi }=-\Psi$ which is N-harmonic in $\mathbb {R}^N_+$ and vanishes on $\partial \mathbb {R}^{N}_+{\setminus }\{0\}$ is transformed by the inversion ${\mathcal I}_{\omega }$ of center $\omega$ and power 1 into the function $\Psi _\omega =\hat{\Psi }\circ {\mathcal I}_{\omega }$ which is positive and N-harmonic in $B^c_{\frac{1}{2}}(a)$ and vanishes on $\partial B_{\frac{1}{2}}(a){\setminus }\{0\}$. For $\epsilon >0$ we denote by $\Phi _\epsilon$ the solution of

$$\begin{aligned} -\Delta _N\Phi _\epsilon&=0\qquad \;\text {in } \Omega \cap B^c_\epsilon \nonumber \\ \Phi _\epsilon&=0\qquad \;\text {in } (B^c_{\frac{1}{2}}(a')\cap \partial B_\epsilon )\cup (\partial \Omega \cap B^c_\epsilon )\nonumber \\ \Phi _\epsilon&=\Psi _{\omega '}\quad \text {in }B_{\frac{1}{2}}(a')\cap \partial B_\epsilon . \end{aligned}$$

(3.60)

If $0<\epsilon '<\epsilon$, $\Phi _{\epsilon '}\ge \Psi _{\omega '}$ in $B_{\frac{1}{2}}(a')\cap \partial B_\epsilon$, thus $\Phi _{\epsilon '}\ge \Phi _{\epsilon '}$ in $\Omega \cap B^c_\epsilon$. We also denote by $\hat{U}_\epsilon$ the solution of

$$\begin{aligned} -\Delta _N\hat{\Phi }_\epsilon&=0\qquad \;\text {in } \Omega \cap B^c_\epsilon \nonumber \\ \hat{\Phi }_\epsilon&=0\qquad \;\text {in } \partial \Omega \cap B^c_\epsilon \nonumber \\ \hat{\Phi }_\epsilon&=\Psi _{\omega }\quad \;\text {in } \Omega \cap \partial B^c_\epsilon . \end{aligned}$$

(3.61)

In the same way as above

$$\begin{aligned} 0<\epsilon '<\epsilon \Longrightarrow \hat{\Phi }_{\epsilon '}\le \hat{\Phi }_\epsilon \quad \text{ in } \,\Omega \cap \partial B^c_\epsilon \end{aligned}$$

Using the explicit form of $\Psi$, ${\mathcal I}_\omega :x\mapsto \omega +\frac{x-\omega }{|x-\omega |^2}$ and ${\mathcal I}_{\omega '}:x\mapsto \omega '+\frac{x-\omega '}{|x-\omega '|^2}$ we see that

thus

$$\begin{aligned} \Phi _\epsilon \le {\frac{1+\epsilon }{1-\epsilon } }\hat{\Phi }_\epsilon \quad \text {in }\Omega \cap B^c_\epsilon . \end{aligned}$$

Letting $\epsilon \rightarrow 0$ we conclude that $\Phi _\epsilon$ converges uniformly in $\overline{\Omega }{\setminus } \{0\}$ to $\Phi _*$ which vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies (3.59). $\square$

The proof of the next statement is similar to the one of Lemma 3.8 up to some minor modifications, so we omit it.

Lemma 3.12

Let the assumptions on q and $\Omega$ of Theorem 3.10 be satisfied. If for some $k>0$ there exists a solution $u_{k}$ of (3.17) in $\Omega$, which belongs to $C^1(\overline{\Omega }{\setminus }\{0\})$, vanishes on $\partial \Omega {\setminus }\{0\}$ and satisfies (3.57), then for any $k>0$ there exists such a solution.

Lemma 3.13

Under the assumptions of Theorem 3.10 there exists a Lipschitz continuous nonnegative subsolution $\tilde{u}$ of (3.17) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$, is smaller than $\Phi _*$ and satisfies

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow 0\\ x/\left| x\right| \rightarrow \sigma \end{array}}\left| x\right| \tilde{u}(x)=\sigma , \end{aligned}$$

(3.62)

uniformly with respect to $\sigma \in S^{N-1}_+$.

Proof

Let $\tau >0$ to be fixed and let w be the solution of

$$\begin{aligned} -\Delta _Nw+|\nabla w|^q=0\quad \text {in }B_2^- \end{aligned}$$

(3.63)

which vanishes on $\partial B_2^-{\setminus }\{0\}$ and satisfies

$$\begin{aligned} \lim _{\begin{array}{c} x\rightarrow 0\\ {x/|x|}\rightarrow \sigma \end{array}}|x|w(x)=\sigma \end{aligned}$$

(3.64)

in the $C^1$-topology of $S^{N-1}_-$. Its existence follows from Theorem 3.6 and this function is dominated by the N-harmonic function $\Phi _*$ corresponding to this domain, obtained in Lemma 3.11. By ${\mathcal I}_{\omega '}$, the half-ball $B_2^-$ is transform into the lunule $G=B_{\frac{1}{2}}(a'){\setminus } B_{\frac{2}{3}}(\frac{4}{3}\omega ')$ and $\tilde{w}=w\circ {\mathcal I}_{\omega '}$ satisfies

$$\begin{aligned} -\Delta _N\tilde{w}+|x-\omega '|^{2(q-N)}|\nabla \tilde{w}|^q=0\quad \text {in }G. \end{aligned}$$

(3.65)

Since $|x-\omega '|\le 1$ in G, $-\Delta _N\tilde{w}+|\nabla \tilde{w}|^q\le 0$ in G. We extend $\tilde{w}$ by 0 in $\Omega {\setminus } G$ and the resulting function $\tilde{u}$ is a subsolution of (3.17) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\})$, is smaller than the N-harmonic function $\Phi _*$ obtained in Lemma 3.11, and satisfies (3.62). $\square$

4 Classification of boundary singularities

We assume that $\Omega \subset \mathbb {R}^N$ is a $C^2$ domain and $0\in \partial \Omega$. Furthermore, in order to avoid extremely technical computations, we shall assume either that $\partial \Omega$ is flat near 0 or $p=N$. We suppose that the tangent plane to $\partial \Omega$ at 0 is $\partial \mathbb {R}^N_+=\{x=(x',0)\}$ and the normal inward unit vector at 0 is $\mathbf{e}_N$, therefore $\mathbf{n}=-\mathbf{e}_N$ in the sequel. We denote by $\omega _{s^{{N-1}}_+}$ the unique positive solution of (3.1) in $S^{N-1}_+$ and by $U_{s^{{N-1}}_+}$ the corresponding singular solution of (1.1) in $\mathbb {R}^N_+$ defined by

$$\begin{aligned} U_{s^{{N-1}}_+}(x)=\left| x\right| ^{-\beta _q}\omega _{s^{{N-1}}_+}\left( \frac{x}{\left| x\right| }\right) . \end{aligned}$$

(4.1)

We recall that ${\psi _*}$ is the unique positive solution of (3.2) with maximum 1 and $\Psi _*$ the corresponding p-harmonic function

$$\begin{aligned} \Psi _*(x)=\left| x\right| ^{-\beta _*}\psi _*\left( \frac{x}{\left| x\right| }\right) . \end{aligned}$$

(4.2)

4.1 The case $1<p<N$

The first statement points out the link between weak and strong singularities.

Proposition 4.1

Under the assumptions of Theorem 3.6 there exists $\lim _{k\rightarrow \infty }u_k=u_\infty$ which is the unique element of $C(\overline{\Omega }{\setminus }\{0\})\cap C^1(\Omega )$ which vanishes on $\partial \Omega {\setminus }\{0\}$, satisfies (1.1) in $\Omega$ and

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{u_{\infty }(x)}{U_{s^{{N-1}}_+}(x)}=1. \end{aligned}$$

(4.3)

Proof

Uniqueness follows from (4.3) and the maximum principle. For existence, since the mapping $k\mapsto u_k$ is increasing and $u_k\le U_{s^{{N-1}}_+}$, there exists $\lim _{k\rightarrow \infty }u_k:=u_\infty \le U_{s^{{N-1}}_+}$ and $u_\infty \in C(\overline{\Omega }{\setminus }\{0\})\cap C^1(\Omega )$. It vanishes on $\partial B^+_\delta {\setminus }\{0\}$ and satisfies (1.1) in $B^+_\delta$. In order to take into account the domain $B^+_\delta$ in the notations, we set $u_k=u_{k,\delta }$. Since the mapping $\delta \mapsto u_{k,\delta }$ is also increasing and $u_{k,\delta }\le k{\Psi _*}$, there also exists $\lim _{\delta \rightarrow \infty }u_{k,\delta }:=u_{k,\infty }\le k{\Psi _*}$ Then, for all $\ell >0$,

$$\begin{aligned} T_\ell [u_{k,\delta }](x)=\ell ^{\beta _q}u_{k,\delta }(\ell x)=u_{k\ell ^{\beta _q},\ell ^{-1}\delta }(x). \end{aligned}$$

(4.4)

Letting $k\rightarrow \infty$, we obtain

$$\begin{aligned} T_\ell [u_{\infty ,\delta }](x)=\ell ^{\beta _q}u_{\infty ,\delta }(\ell x)=u_{\infty ,\ell ^{-1}\delta }(x), \end{aligned}$$

(4.5)

and letting $\delta \rightarrow \infty$, we obtain

$$\begin{aligned} T_\ell [u_{\infty ,\infty }](x)=\ell ^{\beta _q}u_{\infty ,\infty }(\ell x)=u_{\infty ,\infty }(x). \end{aligned}$$

(4.6)

This implies that

$$\begin{aligned} u_{\infty ,\infty }(r,\sigma )=r^{-\beta _q}\omega '(\sigma ), \end{aligned}$$

(4.7)

and $\omega '$ is a positive solution of problem (3.1). Therefore $\omega '=\omega _{s^{{N-1}}_+}$ by Theorem 3.2. If we let $\ell \rightarrow 0$ in (4.4) and take $\left| x\right| =1$, $x=\sigma$, we derive

$$\begin{aligned} \lim _{\ell \rightarrow 0}\ell ^{\beta _q}u_{\infty ,\delta }(\ell , \sigma )=\lim _{\ell \rightarrow 0}u_{\infty ,\ell ^{-1}\delta }(1,\sigma )=u_{\infty ,\infty }(1,\sigma )=\omega _{s^{{N-1}}_+}(\sigma ). \end{aligned}$$

(4.8)

This convergence holds in $C^1(\overline{S^{N-1}_+})$ because of Lemma 2.5. This implies (4.3). $\square$

The main classification result is as follows.

Theorem 4.2

Assume $1<p<N$, $p-1<q<q^*$ and $\partial \Omega \cap B_\delta =\{x=(x',0):\left| x'\right| <\delta \}$, for some $\delta >0$. If $u\in C(\overline{\Omega }{\setminus }\{0\})\cap C^1(\Omega )$ is a positive solution of (1.1) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$, then we have the following alternative:

• (i) either there exists $k\ge 0$ such that

  $$\begin{aligned} \lim _{x\rightarrow 0}\frac{u(x)}{{\Psi _*}(x)}=k, \end{aligned}$$

  (4.9)

• (ii) or

  $$\begin{aligned} \lim _{x\rightarrow 0}\frac{u(x)}{U_{s^{{N-1}}_+}(x)}=1. \end{aligned}$$

  (4.10)

Proof

Step 1. Assume

$$\begin{aligned} \liminf _{x\rightarrow 0}\frac{u(x)}{{\Psi _*}(x)}<\infty , \end{aligned}$$

(4.11)

then we claim that (4.9) holds. We first note that if (4.11) holds, there also holds

$$\begin{aligned} \liminf _{x\rightarrow 0}\frac{u(x)}{u_1(x)}<\infty , \end{aligned}$$

(4.12)

where $u_1$ is the solution of (1.1) obtained in Theorem 3.6 with $k=1$. If $\{x_n\}$ is converging to 0 and such that for some $k>0$

$$\begin{aligned} \liminf _{x\rightarrow 0}\frac{u(x)}{u_1(x)}=k=\lim _{n\rightarrow \infty }\frac{u(x_n)}{u_1(x_n)}, \end{aligned}$$

there also holds by the boundary Harnack inequality (2.38) applied to both u and $u_1$,

$$\begin{aligned} \frac{u(x_n)}{u_1(x_n)}=\frac{u(x_n)}{d(x_n)}\frac{d(x_n)}{u_1(x_n)}\ge c_5^{-2}\frac{u(x)}{u_1(x)}\quad \forall x\text { s.t. }\left| x\right| =\left| x_n\right| . \end{aligned}$$

This implies in particular

$$\begin{aligned} u(x)\le c_5^{2}(k+\epsilon _n)u_1(x)\quad \forall x\text { s.t. }\left| x\right| =\left| x_n\right| \end{aligned}$$

where $\{\epsilon _n\}$ is converging to $0_+$, and by the comparison principle

$$\begin{aligned} u(x)\le Ku_1(x)\quad \forall x\in \mathbb {R}^N_+\text { s.t. }\left| x_n\right| \le \left| x\right| \le \frac{\delta }{2}, \end{aligned}$$

for some $K>0$ and all $n\in \mathbb N_*$. Therefore

$$\begin{aligned} \limsup _{x\rightarrow 0}\frac{u(x)}{u_1(x)}<\infty . \end{aligned}$$

(4.13)

We can assume that $k\ne 0$, otherwise (4.9) holds with $k=0$ and actually u remains bounded near 0. As a consequence of the Hopf Lemma and $C^1$ regularity, there exists $K>0$ such that

$$\begin{aligned} u(x)\le K{\Psi _*}(x)\quad \forall x\in B^+_{\frac{\delta }{2}}. \end{aligned}$$

(4.14)

Let $m=\max \{u(x):\left| x\right| =\delta \}$. For $0<\tau <\delta$ we denote by $k_\tau$ the minimum of the $\kappa >0$ such that $u(x)\le \kappa {\Psi _*}(x)+m$ for $\tau \le \left| x\right| \le \delta$. Then $u(x)\le k_\tau {\Psi _*}(x)+m$, and either the graphs of the mappings $u(\cdot )$ and $k_\tau {\Psi _*}(\cdot )+m$ are tangent at some $x_\tau \in B^+_\delta {\setminus }\overline{B}^+_\tau$, or they are tangent on the boundary of the domain, and the only possibility is that they are tangent on $\left| x\right| =\tau$. Since

$$\begin{aligned} \left| \nabla {\Psi _*}(x)\right| ^2=\left| x\right| ^{-2(\beta _*+1)}(\beta _*^2\psi _*^2+\left| \nabla {\psi _*}\right| ^2), \end{aligned}$$

it never vanishes. If we set $w=u-(k_\tau {\Psi _*}(x)+m)$, then

$$\begin{aligned} -{\mathcal L}w+\left| \nabla u\right| ^q=0 \end{aligned}$$

(4.15)

where the operator

$$\begin{aligned} {\mathcal L}=\sum _{i,j}{\frac{\partial }{\partial x_i} }\left( a_{ij}{\frac{\partial }{\partial x_j} }\right) \end{aligned}$$

is uniformly elliptic in a neighborhood of $x_\tau$ (see [6, Lemma 1.3]). Furthermore $w\le 0$ and $w(x_\tau )=0$ by the strong maximum principle $\nabla u(x_\tau )$ must vanish, which contradicts the fact that $\nabla u(x_\tau )=\nabla w(x_\tau )$ by the tangency condition, and $\nabla w(x_\tau )\ne 0$. Therefore $\left| x_\tau \right| =\tau$ and $x_\tau \notin \partial \mathbb {R}^N_+$. If $\tau '<\tau$, $k_\tau \le k_{\tau '}$, and we set $k=\lim _{\tau \rightarrow 0}k_{\tau }$, which is finite because of (4.14). There exists $\{\tau _n\}$ such that $\sigma _n:=\tau ^{-1}x_{\tau _n}\rightarrow \sigma _0$. Furthermore

$$\begin{aligned} r^{\beta _*}u(r,\sigma )\le k_\tau {\psi _*}(\sigma )+mr^{\beta _*}\quad \text {if }\tau \le r\le \delta \quad \text {and}\quad \tau ^{\beta _*}u(\tau ,\sigma _\tau )= k_\tau {\psi _*}(\sigma _\tau )+m\tau ^{\beta _*}. \end{aligned}$$

(4.16)

Put

$$\begin{aligned} u_\tau (x)=\tau ^{\beta _*}u(\tau x) \end{aligned}$$

(4.17)

Then

$$\begin{aligned} -\Delta _p u_\tau +\tau ^{p-q-\beta _*(p+1-q)}\left| \nabla u_\tau \right| ^q=0\qquad \text {in }B^+_{\frac{\delta }{\tau }}{\setminus } \{0\} \end{aligned}$$

and, by (4.14),

$$\begin{aligned} 0\le u_\tau (x)\le K\left| x\right| ^{-\beta _*}\quad \text {in }B^+_{\frac{\delta }{2\tau }}{\setminus } \{0\}. \end{aligned}$$

By Lemma 2.5, the set of functions $\{u_\tau (\cdot )\}$ is relatively compact in the $C_{loc}^1$ topology of $\overline{\mathbb {R}^N_+}{\setminus } \{0\}$. Therefore, as $q<q^*$, there exist a sequence $\{\tau '_n\}\subset \{\tau _n\}$ converging to 0, and a positive p-harmonic function v in $\mathbb {R}^N_+$, continuous in $\overline{\mathbb {R}^N_+}{\setminus } \{0\}$ and vanishing on $\partial \mathbb {R}^N_+{\setminus } \{0\}$, such that $u_{\tau '_n}\rightarrow v$, and v satisfies (4.14) in $\overline{\mathbb {R}^N_+}{\setminus } \{0\}$. By Theorem 5.1 in Appendix A, there exists $k^*$ such that $v=k^*{\Psi _*}$. In particular,

$$\begin{aligned} \lim _{\tau '_n\rightarrow 0}u_{\tau '_n}(1,\sigma )=k^*{\psi _*}(\sigma ) \end{aligned}$$

(4.18)

in the $C^1(\overline{S^{N-1}_+})$ topology. Combining (4.16), (4.17) and (4.18) we conclude that $k^*=k$ and

$$\begin{aligned} \lim _{\tau '_n\rightarrow 0}\tau '^{\beta _*}_nu_{\tau '_n}(1,\sigma )=k{\psi _*}(\sigma ) \end{aligned}$$

(4.19)

Using Theorem 3.6, it is equivalent to

$$\begin{aligned} \lim _{\tau '_n\rightarrow 0}{\frac{u(\tau '_n,\sigma )}{u_k(\tau '_n,\sigma )} }=1 \end{aligned}$$

(4.20)

uniformly on $S^{N-1}_+$. For any $\epsilon >0$, there exists $n_\epsilon >0$ such that $n\ge n_\epsilon$ implies

$$\begin{aligned} u_{k-\epsilon }(\tau '_n,\sigma )\le u(\tau '_n,\sigma )\le u_{k+\epsilon }(\tau '_n,\sigma ) \end{aligned}$$

By comparison principle,

$$\begin{aligned} u_{k-\epsilon }\le u\le u_{k+\epsilon }+m\quad \text {in }B^+_\delta {\setminus } B^+_{\tau '_n}, \end{aligned}$$

(4.21)

and finally

$$\begin{aligned} u_{k-\epsilon }\le u\le u_{k+\epsilon }+m\quad \text {in }B^+_\delta , \end{aligned}$$

(4.22)

Since $\epsilon$ is arbitrary and using again Theorem 3.6, it implies

$$\begin{aligned} \lim _{r\rightarrow 0}{\frac{u(r,\sigma )}{{\Psi _*}(r,\sigma )} }=k, \end{aligned}$$

(4.23)

locally uniformly on $S^{N-1}$. But since the convergence holds in $C^1(\overline{S^{N-1}_+})$, (4.9) follows.

Step 2. Assume

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{u(x)}{{\Psi _*}(x)}=\infty . \end{aligned}$$

(4.24)

For any $0<\epsilon <\delta$ and $k>0$, there holds

$$\begin{aligned} u_k(x)\le u(x)\le v_\epsilon (x)\quad \text {in }B_\delta ^+{\setminus } B_\epsilon ^+ \end{aligned}$$

(4.25)

where $v_\epsilon$ has been defined in (3.12) and $u_k$ is given by Theorem 3.6. Letting $\epsilon \rightarrow 0$, $k\rightarrow \infty$, and using Proposition 4.1, we derive

$$\begin{aligned} u_\infty (x)\le u(x)\le v_0(x)\quad \text {in }B_\delta ^+{\setminus } \{0\}. \end{aligned}$$

(4.26)

We have seen in Theorem 3.3 that $v_0$ is a separable solution of (1.1) in $\mathbb {R}^N_+$ which vanishes on $\partial \mathbb {R}^N_+{\setminus }\{0\}$, therefore $v_0(x)=U_{s^{{N-1}}_+}(x)$. This implies

$$\begin{aligned} u_\infty (x)\le u(x)\le \left| x\right| ^{-\beta _q}\omega _{s^{{N-1}}_+}\left( \frac{x}{\left| x\right| }\right) \quad \text {in }B_\delta ^+{\setminus } \{0\}. \end{aligned}$$

(4.27)

We conclude using Proposition 4.1. $\square$

4.2 The case $p=N$

When $p=N$, the assumption that $\partial \Omega$ is an hyperplane near 0 can be removed. The proof of the next results is based upon Theorem 3.10. The following result is the extension to the case $p=N$ of Proposition 4.1.

Proposition 4.3

Under the assumptions of Theorem 3.10 there exists $\lim _{k\rightarrow \infty }u_k=u_\infty$ which is the unique element of $C(\overline{\Omega }{\setminus }\{0\})\cap C^1(\Omega )$ which satisfies (3.17) in $\Omega$, vanishes on $\partial \Omega {\setminus }\{0\}$ and such that

$$\begin{aligned} \lim _{x\rightarrow 0}\frac{u_{\infty }(x)}{U_{s^{{N-1}}_+}(x)}=1. \end{aligned}$$

(4.28)

Proof

We denote by $u_k^{\Omega }$ the unique positive solution of (3.17) satisfying (3.57) obtained in Theorem 3.6. Then

$$\begin{aligned} T_\ell [u_k^\Omega ]=u^{\Omega ^{\ell }}_{\ell ^{\beta _q-\beta _*}k}, \end{aligned}$$

(4.29)

because of uniqueness. We denote by $B:=B_{\frac{1}{2}}(a)$ and $B':=B_{\frac{1}{2}}(a')$ the two balls tangent to $\partial \Omega$ at 0 respectively interior and exterior to $\Omega$ introduced in the proof of Lemma 3.11. Estimate (3.58) implies

$$\begin{aligned} u_k^{B'^c}\le u_k^\Omega \le u_k^B \end{aligned}$$

(4.30)

the left-hand side inequality holding in $\Omega$ and the right-hand side one in B. Therefore

$$\begin{aligned} T_\ell [u_k^{B'^c}]:=u^{B'^{c\,\ell }}_{\ell ^{\beta _q-\beta _*}k}\le T_\ell [u_k^\Omega ]\le T_\ell [u_k^B]:=u^{B^{\ell }}_{\ell ^{\beta _q-\beta _*}k}, \end{aligned}$$

(4.31)

the domains of validity of these inequalities being modified accordingly. Using again (3.58) we obtain

$$\begin{aligned} T_{\ell '}[u_{k'}^{B'^c}]\le T_\ell [u_k^{B'^c}]\quad \text {in } B'^{c\,\ell '}, \end{aligned}$$

(4.32)

for any $0<\ell '\le \ell$ and $\ell '^{\beta _q-\beta _*}k'\le \ell ^{\beta _q-\beta _*}k$. In the same way

$$\begin{aligned} T_{\ell '}[u_{k'}^{B}]\ge T_\ell [u_k^{B}]\quad \text {in } B^{\ell }, \end{aligned}$$

(4.33)

for any $0<\ell '\le \ell$ and $\ell '^{\beta _q-\beta _*}k'\ge \ell ^{\beta _q-\beta _*}k$. Since $u_k^\Omega$ $u_k^B$, $u_k^{B'^c}$ are increasing with respect to k, they converge respectively to $u_\infty ^\Omega$ $u_\infty ^B$, $u_\infty ^{B'^c}$ and there holds for any $\ell >0$

$$\begin{aligned} T_\ell [u_\infty ^{B'^c}]\le T_\ell [u_\infty ^\Omega ]\le T_\ell [u_\infty ^{B}], \end{aligned}$$

(4.34)

from (4.31) and

$$\begin{aligned} \begin{array}{ll} (i)\quad T_{\ell '}[u_{\infty }^{B'^c}]\le T_\ell [u_\infty ^{B'^c}]&{}\quad \text {in } B'^{c\,\ell '}\\ (ii)\quad T_{\ell '}[u_{\infty }^{B}]\ge T_\ell [u_\infty ^{B}]&{}\quad \text {in } B^{\ell } \end{array} \end{aligned}$$

(4.35)

for any $0<\ell '\le \ell$. Notice that , replacing $\ell$ by $\ell \ell '$ we can rewrite (4.34) as follows

$$\begin{aligned} T_{\ell '}[T_\ell [u_\infty ^{B'^c}]]\le T_{\ell '}[T_\ell [u_\infty ^\Omega ]]\le T_{\ell '}[T_\ell [u_\infty ^{B}]]. \end{aligned}$$

(4.36)

Because of the monotonicity with respect to $\ell$ the following limits exist

$$\begin{aligned} U^{B'^c}=\lim _{\ell \rightarrow 0}T_\ell [u_\infty ^{B'^c}]\quad \text {and}\quad U^{B}=\lim _{\ell \rightarrow 0}T_\ell [u_\infty ^{B}]. \end{aligned}$$

(4.37)

By Lemma 2.5 applied with $\phi (|x|)=|x|^{-\beta _q}$ and since there holds $u_\infty ^B(x)\le c|x|^{-\beta _q}$ and $u_\infty ^{B'}(x)\le c|x|^{-\beta _q}$, we derive

$$\begin{aligned} \begin{array}{ll} (i)\quad |\nabla T_\ell [u_\infty ^B](x)|\le c_2|x|^{-\beta _q-1}&{}\quad \forall x\in B^{\ell }\\ (ii)\quad |\nabla T_\ell [u_\infty ^B](x)-\nabla T_\ell [u_\infty ^B](y)|\le c_2|x|^{-\beta _q-1-\alpha }|x-y|^{\alpha }&{}\quad \forall x,y\in B^{\ell },\;|x|\le |y|\\ (iii) \quad T_\ell [u_\infty ^B](x)\le c_2|x|^{-\beta _q-1} (\text{ dist }\,(x,\partial B^\ell ))^{\alpha }&{}\quad \forall x\in B^{\ell }, \end{array} \end{aligned}$$

(4.38)

and

$$\begin{aligned} \begin{array}{lll} (i)\quad |\nabla T_\ell [u_\infty ^{B'^c}](x)|\le c_2|x|^{-\beta _q-1}&{}\quad \forall x\in B'^{c\,\ell }\\ (ii) \quad |\nabla T_\ell [u_\infty ^{B'^c}](x)-\nabla T_\ell [u_\infty ^{B'^c}](y)|\le c_2|x|^{-\beta _q-1-\alpha }|x-y|^{\alpha }&{}\quad \forall x,y\in B'^{c\,\ell },\;|x|\le |y|\\ (iii)\quad T_\ell [u_\infty ^{B'^c}](x)\le c_2|x|^{-\beta _q-1} (\text{ dist }\,(x,\partial B'^{c\,\ell }))^{\alpha }&{}\quad \forall x\in B'^{c\,\ell }. \end{array} \end{aligned}$$

(4.39)

Thus the sets of functions $\{T_\ell [u_\infty ^B]\}$ and $\{T_\ell [u_\infty ^{B'}]\}$ are equicontinuous in the $C^1$-loc topology and by uniqueness, the limit in (4.37) below holds in this topology. Hence $U^{B'^c}$ and $U^{B^c}$ are positive solutions of (3.17) in $\mathbb {R}^N_+$ which vanish on $\partial \mathbb {R}^N_+{\setminus }\{0\}$. Furthermore $U^{B'^c}\le U^{B^c}$ Since for any $\ell ,\ell '>0$, $T_{\ell '}[T_\ell [u_\infty ^{B'^c}]]=T_{\ell \ell '}[u_\infty ^{B'^c}]$, it follows $T_{\ell '}[U^{B'^c}]=U^{B'^c}$ and in the same way $T_{\ell '}[U^{B}]=U^{B}$. This means that $U^{B}$ and $U^{B'^c}$ are self-similar solutions of (3.17) in $\mathbb {R}^N_+$ and they vanish on $\partial \mathbb {R}^N_+{\setminus }\{0\}$. Hence

$$\begin{aligned} U^{B}=U^{B'^c}=U_{S^{N-1}_+}. \end{aligned}$$

(4.40)

Applying again Lemma 2.5 to $u_\infty ^\Omega$ with $\phi (|x|)=|x|^{-\beta _q}$ we have

$$\begin{aligned} \begin{array}{ll} (i)\quad |\nabla T_\ell [u_\infty ^\Omega ](x)|\le c_2|x|^{-\beta _q-1}&{}\quad \forall x\in \Omega ^{\ell }\\ (ii)\quad |\nabla T_\ell [u_\infty ^\Omega ](x)-\nabla T_\ell [u_k^\Omega ](y)|\le c_2|x|^{-\beta _q-1-\alpha }|x-y|^{\alpha }&{}\quad \forall x,y\in \Omega ^{\ell },\;|x|\le |y|\\ (iii)\quad T_\ell [u_\infty ^\Omega ](x)\le c_2|x|^{-\beta _q-1} (\text{ dist }\,(x,\partial \Omega ^\ell ))^{\alpha }&{}\quad \forall x\in \Omega ^{\ell }. \end{array} \end{aligned}$$

(4.41)

This implies that the set of functions $\{T_\ell [u_\infty ^\Omega ]\}_\ell$ is equicontinuous in the $C^1$-loc topology of $\mathbb {R}^N_+$ and there exists a sequence $\{\ell _n\}\rightarrow 0$ and a function U such that $T_{\ell _n}[u_\infty ^\Omega ]\rightarrow U^\Omega$ in this topology of $\mathbb {R}^N_+$, and U is a positive solution of (3.17) in $\mathbb {R}^N_+$ which vanishes on $\partial \mathbb {R}^N_+{\setminus }\{0\}$. From (4.34) and (4.40) there holds $U^\Omega = U_{S^{N-1}_+}$ and therefore

$$\begin{aligned} \lim _{\ell \rightarrow 0}T_\ell [u_\infty ^\Omega ]=U_{S^{N-1}_+}. \end{aligned}$$

(4.42)

This implies (4.28) and

$$\begin{aligned} \lim _{r\rightarrow 0}r^{\beta _q}u_\infty ^\Omega (r,\sigma )=\omega _{S^{N-1}_+}(\sigma ) \end{aligned}$$

(4.43)

uniformly on compact subsets of $S^{N-1}_+$. $\square$

Up to minor modifications the proof of the next classification theorem is similar to the one of Theorem 4.2.

Theorem 4.4

Assume $N-1<q<N-\frac{1}{2}$ If $u\in C(\overline{\Omega }{\setminus }\{0\})\cap C^1(\Omega )$ is a positive solution of (3.17) in $\Omega$ which vanishes on $\partial \Omega {\setminus }\{0\}$, then we have the following alternative:

• (i) either there exists $k\ge 0$ such that (4.9) holds,

• (ii) or (4.10) holds.

Appendices

Appendix A: Positive p-harmonic functions in a half space

In this section we prove the following rigidity result.

Theorem 5.1

Assume $1<p\le N$ and $u\in C^1(\mathbb {R}^N_+)\cap C(\overline{\mathbb {R}^N_+}{\setminus }\{0\})$ is a positive p-harmonic function which vanishes on $\partial \mathbb {R}_+^N{\setminus }\{0\}$ and such that $\left| x\right| ^{\beta _*} u(x)$ is bounded. Then there exists $k\ge 0$ such that

$$\begin{aligned} u(x)=k{\Psi _*}(x)\quad \forall x\in \mathbb {R}^N_+. \end{aligned}$$

(5.1)

Proof

Since $\left| x\right| ^{\beta _*} u(x)$ is bounded, $\left| x\right| ^{\beta _*+1} \nabla u(x)$ is also bounded and there exists $m>0$ such that $u(x)\le m{\Psi _*}(x)$ in $B_\delta ^+$. We denote by k the infimum of the $c>0$ such that $u(x)\le c\Psi _*(x)$. Then

$$\begin{aligned} 0\le u(x)\le k{\Psi _*}(x)\quad \forall x\in \mathbb {R}^N_+{\setminus }\{0\} \end{aligned}$$

(5.2)

and we assume that $k>0$ otherwise $u=0$. Assume that the graphs over $\mathbb {R}^N_{+}$ of the functions $x\mapsto u(x)$ and $x\mapsto k\Psi _*(x)$ are tangent at some point $x_0\in \mathbb {R}^N_+$ or $x_0\in \partial \mathbb {R}^N_+{\setminus }\{0\}$. Since $\nabla \Psi _*$ never vanishes in $\overline{\mathbb {R}}^N_+{\setminus }\{0\}$ it follows from the strong maximum principle or Hopf Lemma that $u=k{\Psi _*}$. If the two graphs are not tangent in $\overline{\mathbb {R}}^N_+{\setminus }\{0\}$, either they are asymptotically tangent at 0, or at $\infty$.

• (i) In the first case there exists two sequences $\{k_n\}$ increasing to k and $\{x_n\}\subset \mathbb {R}^N_+$ converging to zero such that $\frac{u(x_n)}{{\Psi _*}(x_n)}=k_n$. We set $r_n=\left| x_n\right|$ and $u_{r_n}(x)=r_n^{\beta _*} u(r_nx)$. Then $u_{r_n}$ is p-harmonic and positive and $0<u_{r_n}(x)\le k\left| x\right| ^{-\beta _*}{\psi _*}(\frac{x}{\left| x\right| })$; therefore

  $$\begin{aligned} \left| \nabla u_{r_n}(x)\right| \le C\left| x\right| ^{-\beta _*-1}\quad \text {and}\quad \left| \nabla u_{r_n}(x)-\nabla u_{r_n}(x')\right| \le C\left| x\right| ^{-\beta _*-1-\alpha } \left| x-x'\right| ^\alpha \end{aligned}$$

  (5.3)

  for $0<\left| x\right| \le \left| x'\right|$ and some constants $C>0$ and $\alpha \in (0,1)$. Up to a subsequence, we can assume that $u_{r_n}$ converges to some U in the $C^1_{loc}$ topology of $\overline{\mathbb {R}}^N_+{\setminus }\{0\}$ and $\frac{x_n}{r_n}\rightarrow \xi \in S^{N-1}_+$. The function U is p-harmonic and positive in $\mathbb {R}^N_+$ and satisfies $0\le U\le k{\Psi _*}$ in $\mathbb {R}^N_+$ and $U(\xi )= k{\Psi _*}(\xi )$ if $\xi \in S^{N-1}_+$ or $U_{x_N}(\xi )= k\Psi _{*\,x_N}(\xi )$ if $\xi \in \partial S^{N-1}_+$. It follows from the strong maximum principle or Hopf Lemma that $U= k{\Psi _*}$. Therefore $u_{r_n}\rightarrow k{\Psi _*}$ and in particular

  $$\begin{aligned} \lim _{r_n\rightarrow 0}{\frac{r_n^{\beta _*} u(r_n,\sigma )}{{\psi _*}(\sigma )} }=k\quad \text {uniformly on }\,S^{N-1}_+. \end{aligned}$$

  (5.4)

  For any $\epsilon >0$, there exists $n_\epsilon \in \mathbb N_*$ such that for $n\ge n_\epsilon$, $(k-\epsilon ){\Psi _*}(x)\le u(x)\le (k+\epsilon ){\Psi _*}(x)$ if $\left| x\right| =r_n$. This implies $(k-\epsilon ){\Psi _*}(x)\le u(x)\le (k+\epsilon ){\Psi _*}$ for $\left| x\right| \ge r_n$ and therefore in $\mathbb {R}^N$. Since $\epsilon$ is arbitrary, we deduce that $u=k{\Psi _*}$.

• (ii) if the two graphs are tangent at infinity, there exist two sequences $\{k_n\}$ increasing to k and $\{x_n\}$ such that $r_n=\left| x_n\right| \rightarrow \infty$ with $u(x_n)=k_n{\Psi _*}(x_n)$ and

  $$\begin{aligned} \lim _{r_n\rightarrow \infty }{\frac{r_n^{\beta _*} u(r_n,\sigma )}{{\psi _*}(\sigma )} }=k\quad \text {uniformly on }\,S^{N-1}_+. \end{aligned}$$

  (5.5)

Therefore we look at the supremum of the $c>0$ such that $u\ge c{\Psi _*}$. If the set of such c is empty, it would mean that

$$\begin{aligned} \inf _{x\in \mathbb {R}^N_+}\frac{u(x)}{{\Psi _*}(x)}=0. \end{aligned}$$

Clearly, if this infimum is achieved at some point, the strong maximum principle or Hopf Lemma imply $u\equiv 0$, contradicting (5.5), and this relation prevents also this infimum be achieved at infinity. We are left with the case where there exists a sequence $\{z_n\}\subset \mathbb {R}^N_+$, converging to 0, such that

$$\begin{aligned} \lim _{n\rightarrow \infty }{\frac{u(z_n)}{{\Psi _*}(z_n)} }=0. \end{aligned}$$

(5.6)

By boundary Harnack inequality [2, th 2.11], there exists $c>0$ such that

$$\begin{aligned} c^{-1}{\frac{u(z)}{{\Psi _*}(z)} }\le {\frac{u(z_n)}{{\Psi _*}(z_n)} }\le c{\frac{u(z)}{{\Psi _*}(z)} }\quad \forall z\in \mathbb {R}^N_+\text { s.t. }\left| z\right| =\left| z_n\right| \end{aligned}$$

(5.7)

Combining (5.6) and (5.7), we derive that

$$\begin{aligned} \lim _{n\rightarrow \infty }\sup _{\left| z\right| =\left| z_n\right| }{\frac{u(z)}{{\Psi _*}(z)} }=0, \end{aligned}$$

(5.8)

Denoting by $\epsilon _n$ the supremum in the above relation, we obtain that $u\le \epsilon _n{\Psi _*}$ in $\mathbb {R}^N_+{\setminus } B_{\epsilon _n}$ and finally $u=0$, contradiction. Thus we are left with the case where there exists $k'\in (0,k]$ which is the supremum of the $c>0$ such that $u\ge c{\Psi _*}$. In particular $u\ge k'{\Psi _*}$. Remembering that $u\le k{\Psi _*}$ we get $k=k'$, which implies $u=k{\Psi _*}$.

Next we assume that $k'<k$. Clearly the graphs of u and $k'{\Psi _*}$ cannot be tangent in $\overline{\mathbb {R}}^N_+$, because of strong maximum principle or Hopf Lemma. They cannot be tangent at infinity because of (5.5). Therefore there exist two sequences $\{k'_n\}$ increasing to $k'$ and $\{x'_n\}\subset \mathbb {R}^N_+$ converging to 0 such that $\frac{u(x'_n)}{{\Psi _*}(x'_n)}=k'_n$. As in case (i) we obtain that

$$\begin{aligned} \lim _{r'_n\rightarrow 0}{\frac{r_n'^{\beta _*} u(r'_n,\sigma )}{{\psi _*}(\sigma )} }=k'\quad \text {uniformly on }\,S^{N-1}_+, \end{aligned}$$

(5.9)

where $r'_n=\left| x'_n\right|$, and finally derive that $u=k'{\Psi _*}$, a contradiction with (5.5). Therefore $k=k'$, which ends the proof. $\square$

Remark

In the case $p=N$ the result holds under the weaker assumption $\displaystyle \lim _{\left| x\right| \rightarrow \infty }u(x)=0$. This is due to the fact that this condition implies by regularity

$$\begin{aligned} \lim _{\left| x\right| \rightarrow \infty }{\frac{u(x)}{\omega _{s^{{N-1}}_+}\left( \frac{x}{\left| x\right| }\right) } }=0 \end{aligned}$$

and therefore

$$\begin{aligned} u(x)\le m{\Psi _*}(x)\quad \forall x\,\text { s.t. }\left| x\right| \ge 1, \end{aligned}$$

where $m=\max _{\left| x\right| =1}{\frac{u(x)}{\omega _{s^{{N-1}}_+}(\frac{x}{\left| x\right| })} }$. Using the inversion $x\mapsto \frac{x}{\left| x\right| ^2}$, we obtain that the estimate $u\le m{\Psi _*}$ holds $\mathbb {R}^N$, and we conclude by Theorem 5.1.

Remark

We conjecture that the rigidity result holds under the mere condition

$$\begin{aligned} \lim _{\left| x\right| \rightarrow \infty }\left| x\right| ^{- \tilde{\beta }}u(x)=0, \end{aligned}$$

(5.10)

were $\tilde{\beta }$ is the (positive) exponent corresponding to the regular spherical p-harmonic function under the form

$$\begin{aligned} \tilde{\Psi }=\left| x\right| ^{\tilde{\beta }} {\tilde{\psi }}(\small {\frac{x}{\left| x\right| }}), \end{aligned}$$

(5.10)

see [12, 14]. Note that $\tilde{\beta }=1$ when $p=N$.

Appendix B: Estimates on $\beta _*$

When $N=2$ and $1<p\le 2$, it is proved in [9] that

$$\begin{aligned} \beta _*={\frac{3-p+2\sqrt{p^2-5p+7}}{3(p-1)} }. \end{aligned}$$

(6.1)

Up to now no estimate is known when $N>2$ except in the cases $p=2$ where $\beta _*=N-1$ and $p=N$ where $\beta _*=1$, besides the classical one

$$\begin{aligned} \beta _*>{\frac{N-p}{p-1} }, \end{aligned}$$

(6.2)

valid when $p<N$. In this section we prove the following result

Theorem 6.1

Assume $1<p< N$. Then the following estimates hold:

$$\begin{aligned} 1<p<2\Longrightarrow \beta _*> \frac{N-1}{p-1}, \end{aligned}$$

(6.3)

$$\begin{aligned} 2<p<N\Longrightarrow \max \left\{ 1, \frac{N-p}{p-1}\right\} <\beta _*< \frac{N-1}{p-1}. \end{aligned}$$

(6.4)

Remark

It is worth noticing that when $p=2$ or $p=N$, there holds $\beta _*=\frac{N-1}{p-1}$.

.

Proof of Theorem 6.1

We consider the following set of spherical coordinates in $\mathbb {R}^N_+$ with $x=(x_1,\ldots ,x_N)$

$$\begin{aligned}&x_1=r\sin \theta _{N-1}\sin \theta _{N-2}\ldots \sin \theta _{2}\sin \theta _{1}\nonumber \\&x_2=r\sin \theta _{N-1}\sin \theta _{N-2}\ldots \sin \theta _{2}\cos \theta _{1}\nonumber \\&\vdots \nonumber \\&x_{N-1} =r\sin \theta _{N-1}\cos \theta _{N-2}\nonumber \\&x_N =r\cos \theta _{N-1} \end{aligned}$$

(6.5)

with $\theta _1\in [0,2\pi ]$ and $\theta _k\in [0,\pi ]$ for $k=2,\ldots ,N-2$ and $\theta _{N-1}\in [0,\frac{\pi }{2}]$. Under this representation, a solution $\omega$ of (3.2) verifies

$$\begin{aligned}&-{\frac{1}{\sin ^{N-2}\theta _{N-1}} }\left[ \sin ^{N-2}\theta _{N-1}\left( \beta ^2_*\omega ^2+\omega ^2_{\theta _{N-1}}+{\frac{1}{\sin ^{2}\theta _{N-1}} }\left| \nabla _{\theta '}\omega \right| ^2\right) ^{\frac{p-2}{2}}\omega _{\theta _{N-1}}\right] _{\theta _{N-1}}\nonumber \\&\qquad -{\frac{1}{\sin ^{2}\theta _{N-1}} }div'_{\theta '}\left[ \sin ^{N-2}\theta _{N-1}\left( \beta ^2_*\omega ^2+\omega ^2_{\theta _{N-1}}+{\frac{1}{\sin ^{2}\theta _{N-1}} }\left| \nabla _{\theta '}\omega \right| ^2\right) ^{\frac{p-2}{2}}\nabla _{\theta '}\omega \right] \nonumber \\&\quad =\beta _*\Lambda _{\beta _*}\left[ \sin ^{N-2}\theta _{N-1}\left( \beta ^2_*\omega ^2+\omega ^2_{\theta _{N-1}}+{\frac{1}{\sin ^{2}\theta _{N-1}} }\left| \nabla _{\theta '}\omega \right| ^2\right) ^{\frac{p-2}{2}}\omega \right] \end{aligned}$$

(6.6)

where $\nabla _{\theta '}$ and $div'_{\theta '}$ denotes respectively the spherical gradient the divergence in variables $\theta '=(\theta _1,\ldots ,\theta _{N-2})$ parameterizing $S^{N-2}$ and $\Lambda _{\beta _*}$ is defined in Introduction. If $\omega$ is the unique positive solution of (3.2) (up to homothety), it depends only on $\theta _{N-1}$ and is $C^\infty$. For simplicity we set $\theta _{N-1}=\theta \in [0,\frac{\pi }{2}]$ and $\omega =\omega (\theta )$ satisfies

$$\begin{aligned}&-{\frac{1}{\sin ^{N-2}\theta } }\left[ \sin ^{N-2}\theta \left( \beta ^2_*\omega ^2+\omega ^2_{\theta }\right) ^{\frac{p-2}{2}}\omega _{\theta }\right] _{\theta } =\beta _*\Lambda _{\beta _*}\left[ \sin ^{N-2}\theta \left( \beta ^2_*\omega ^2+\omega ^2_{\theta }\right) ^{\frac{p-2}{2}}\omega \right] \nonumber \\&\quad \text {in }\left( 0,\frac{\pi }{2}\right) \nonumber \\&\quad \omega \left( \frac{\pi }{2}\right) =0,\;\omega _\theta (0)=0. \end{aligned}$$

(6.7)

Step 1: the eigenvalue identity Equation (6.7) can also be written under the form

$$\begin{aligned} -\omega _{\theta \theta }-(N-2)\cot \theta \,\omega _{\theta }-(p-2){\frac{\beta _*^2\omega +\omega _{\theta \theta }}{\beta ^2_*\omega ^2+\omega ^2_{\theta }} }\omega ^2_{\theta } =\beta _*\Lambda _{\beta _*}\omega . \end{aligned}$$

(6.8)

By multiplying (6.8) by $\cos \theta \sin ^{N-2}\theta$ and then integrating over $(0,\frac{\pi }{2})$ we obtain

$$\begin{aligned} - {\int _{0}^{\frac{\pi }{2}}}\left( \omega _{\theta \theta }+(N-2)\cot \theta \,\omega _{\theta }\right) \cos \theta \sin ^{N-2}\theta d\theta =(N-1) {\int _{0}^{\frac{\pi }{2}}}\omega \cos \theta \sin ^{N-2}\theta d\theta . \end{aligned}$$

Noticing that

$$\begin{aligned} \beta _*\Lambda _{\beta _*}+1-N=(p-1)\left( \beta _*-\frac{N-1}{p-1}\right) \left( \beta _*+1\right) \end{aligned}$$

we derive

$$\begin{aligned}&(2-p){\int _{0}^{\frac{\pi }{2}}}{\frac{\beta _*^2\omega +\omega _{\theta \theta }}{\beta ^2_*\omega ^2+\omega ^2_{\theta }} }\omega ^2_{\theta } \omega \cos \theta \sin ^{N-2}\theta d\theta \nonumber \\&\quad =(p-1)\left( \beta _*-{\frac{N-1}{p-1} }\right) \left( \beta _*+1\right) {\int _{0}^{\frac{\pi }{2}}}\omega \cos \theta \sin ^{N-2}\theta d\theta . \end{aligned}$$

(6.9)

Step 2: elliptic coordinates and reduction Writing $\omega (\theta )=\omega (0)+a\theta ^2+o(\theta ^2)$, $\omega _\theta (\theta )=2a\theta +o(\theta )$ and $\omega _{\theta \theta }(\theta )=2a+o(1)$, then $-Na=\beta _*\Lambda _{\beta _*}$. This implies that $\omega$ is decreasing near 0. It is immediate that it cannot have a local minimum in $(0,\frac{\pi }{2})$, therefore it remains decreasing in the whole interval. We parameterize the ellipse

$$\begin{aligned} E_r=\{(x,y): x>0,\,y<0,\,x^2+\beta _*^{-2}y^2=r^2\} \end{aligned}$$

by setting

$$\begin{aligned} \omega =r\cos \phi \quad \text {and}\quad -\omega _\theta =\beta _* r\sin \phi \quad \text {with}\quad \phi =\phi (\theta )\,\text { and }\;r=r(\theta ). \end{aligned}$$

The functions r and $\phi$ are $C^2$. Hence $r_\theta \cos \phi -r\sin \phi \phi _\theta =-\beta _* r\sin \phi$, then $r_\theta \cos \phi =(\phi _\theta -\beta _*)r\sin \phi$ and $r_\theta =(\phi _\theta -\beta _*)r\tan \phi .$ Plugging this into (6.8), we derive

$$\begin{aligned} -\left( (p-1){\frac{r_\theta }{r} }+\phi _\theta \cot \phi +(N-2)\cot \theta \right) +\Lambda _{\beta _*}\cot \phi =0, \end{aligned}$$

(6.10)

and finally

$$\begin{aligned} (p-1)(\phi _\theta -\beta _*)\tan \phi +(\phi _\theta -\Lambda _{\beta _*})\cot \phi =(2-N)\cot \theta . \end{aligned}$$

(6.11)

Step 3: estimates on $\phi _\theta$ We can write (6.11) under the equivalent form

$$\begin{aligned} (p-1)(\phi _\theta -\beta _*)\tan ^2\phi +\phi _\theta -\Lambda _{\beta _*}=(2-N){\frac{\cos \theta }{\cos \phi } }{\frac{\sin \phi }{\sin \theta } }. \end{aligned}$$

(6.12)

Since

$$\begin{aligned} \lim _{\theta \rightarrow 0}{\frac{\sin \phi }{\sin \theta } }=\lim _{\theta \rightarrow 0}{\frac{\cos \phi }{\cos \theta } }\phi _\theta =\phi _\theta (0), \end{aligned}$$

we derive $\phi _\theta (0)-\Lambda _{\beta _*}=(2-N)\phi _\theta (0)$ and thus $\phi _\theta (0)={\frac{\Lambda _{\beta _*}}{N-1} }$. Similarly, the expansion of $\phi (\theta )$ near $\theta =\frac{\pi }{2}$ yields to $\phi _\theta (\frac{\pi }{2})=\beta _*.$ Since $p<N$, $\Lambda _{\beta _*}/(N-1)<\beta _*$. We claim now that

$$\begin{aligned} \phi _\theta (\theta )\le \beta _*\quad \forall \theta \in \left( 0,\frac{\pi }{2}\right) . \end{aligned}$$

(6.13)

If $\Lambda _{\beta _*}\le \beta _*$, then

$$\begin{aligned} (2-N)\cot \theta&=(p-1)(\phi _\theta -\beta _*)\tan \phi +(\phi _\theta -\Lambda _{\beta _*})\cot \phi \\&\ge ((p-1)\tan \phi +\cot \phi )(\phi _\theta -\beta _*) \end{aligned}$$

thus (6.13) holds.

Next we assume $\beta _*< \Lambda _{\beta _*}$. It means $0<(p-2)\beta _*-(N-p)$ and thus $p>2$. We claim that

$$\begin{aligned} \beta _*\le {\frac{N-2}{p-2} }. \end{aligned}$$

(6.14)

We proceed by contradiction and assume

$$\begin{aligned} \beta _*>{\frac{N-2}{p-2} }. \end{aligned}$$

(6.15)

Then

$$\begin{aligned} (p-2)\left( \beta _*^2-\frac{N-p}{p-2}\beta _*-\frac{N-2}{p-2}\right) =(p-2)\left( \beta _*+1\right) \left( \beta _*-\frac{N-2}{p-2}\right) >0. \end{aligned}$$

Equivalently

$$\begin{aligned} \beta _*(\Lambda _{\beta _*}-\beta _*)>N-2. \end{aligned}$$

Since

$$\begin{aligned} \lim _{\theta \rightarrow \frac{\pi }{2}}\cot \theta \tan \phi =\lim _{\theta \rightarrow \frac{\pi }{2}}\frac{\cos \theta }{\cos \phi }=\lim _{\theta \rightarrow \frac{\pi }{2}}\frac{\sin \theta }{\phi _\theta \sin \phi }=\frac{1}{\beta _*} \end{aligned}$$

and

$$\begin{aligned} (p-1)(\phi _\theta (\theta )-\beta _*)\tan ^2\phi&=\Lambda _{\beta _*}-\phi _\theta (\theta )+(2-N){\frac{\cos \theta }{\cos \phi } }{\frac{\sin \phi }{\sin \theta } }\nonumber \\&={\frac{1}{\beta _*} }\left( \beta _*(\Lambda _{\beta _*}-\beta _*)+2-N\right) +o(1), \end{aligned}$$

(6.16)

thus, if (6.15) holds there exists $\epsilon >0$ such that $\phi _\theta (\theta )>\beta _*$ for any $\theta \in [\frac{\pi }{2}-\epsilon ,\frac{\pi }{2})$. Since $\phi _\theta (0)<\beta _*$, there exists $\bar{\theta }\in (0,\frac{\pi }{2})$ such that $\phi _\theta (\bar{\theta })=\beta _*$ and $\phi _{\theta \theta }(\bar{\theta })\ge 0$. We compute $\phi _{\theta \theta }$ and get

$$\begin{aligned}&(p-1)\phi _\theta (\phi _\theta -\beta _*)\sec ^2\phi +\left( (p-1)\tan \phi +\cot \phi \right) \phi _{\theta \theta } -\phi _\theta (\phi _\theta -\Lambda _{\beta _*})\csc ^2\phi \\&\quad =(N-2)\csc ^2\theta \end{aligned}$$

Hence, at $\theta =\bar{\theta }$

$$\begin{aligned} \phi _{\theta \theta }(\bar{\theta })\left( (p-1)\tan \phi (\bar{\theta })+\cot \phi (\bar{\theta })\right) = \beta _*(\beta _*-\Lambda _{\beta _*})\csc ^2\phi (\theta )+(N-2)\csc ^2\bar{\theta }\end{aligned}$$

From (6.11),

$$\begin{aligned} \cot \phi (\bar{\theta })=\frac{N-2}{\Lambda _{\beta _*}-\beta _*}\cot \bar{\theta }\end{aligned}$$

Therefore

$$\begin{aligned} A(\bar{\theta })&:=\phi _{\theta \theta }(\bar{\theta })\left( (p-1)\tan \phi (\bar{\theta })+\cot \phi (\bar{\theta })\right) \nonumber \\&= \left( 1+\left( {\frac{N-2}{\Lambda _{\beta _*}-\beta _*} }\right) ^2\cot ^2\bar{\theta }\right) \beta _*(\beta _*-\Lambda _{\beta _*})+(N-2)(1+\cot ^2\bar{\theta })\nonumber \\&=\beta _*(\beta _*-\Lambda _{\beta _*})+N-2-\left( {\frac{(N-2)^2}{\Lambda _{\beta _*}-\beta _*} }+2-N\right) \cot ^2\bar{\theta }\nonumber \\&=-(p-2)(\beta _*+1)\left( \beta _*-{\frac{N-2}{p-2} }\right) -{\frac{N-2}{\Lambda _{\beta _*}-\beta _*} }\left( \beta _*(N-1)-\Lambda _{\beta _*}\right) \cot ^2\bar{\theta }\nonumber \\&<0, \end{aligned}$$

(6.17)

using (6.15) and the fact that $N>p$. This is a contradiction, thus (6.14) holds.

Next, if $\beta _*< \frac{N-2}{p-2}$, it follows from (6.16) that there exists $\epsilon >0$ such that $\phi _\theta <\beta _*$ in $[\frac{\pi }{2}-\epsilon ,\frac{\pi }{2})$. If (6.13) is not true, there exist $0<\theta _1<\theta _2<\frac{\pi }{2}-\epsilon$ such that $\phi _\theta (\theta _1)=\phi _\theta (\theta _2)=\beta _*$, $\phi _{\theta \theta }(\theta _1)\ge 0$, $\phi _{\theta \theta }(\theta _2)\le 0$. Using the equation satisfied by $\phi _{\theta \theta }$, we obtain for $i=1,2$,

$$\begin{aligned} A(\theta _i) =(2-p)(\beta _*+1)\left( \beta _*-{\frac{N-2}{p-2} }\right) -{\frac{N-2}{\Lambda _{\beta _*}-\beta _*} }(\beta _*(N-1)-\Lambda _{\beta _*})\cot ^2\theta _i.\qquad \quad \end{aligned}$$

(6.18)

On one hand $A(\theta _2)\le 0\le A(\theta _1)$, and on the other

$$\begin{aligned} A(\theta _2)- A(\theta _1)={\frac{N-2}{\Lambda _{\beta _*}-\beta _*} }(\beta _*(N-1)-\Lambda _{\beta _*})(\cot ^2\theta _1-\cot ^2\theta _2)>0, \end{aligned}$$

since $\cot$ is decreasing in $(0,\frac{\pi }{2})$, $\cot ^2\theta _1>\cot ^2\theta _2$, a contradiction. Therefore $\phi _\theta \le \beta _*$ in $(0,\frac{\pi }{2})$.

Finally, if $\beta _*= \frac{N-2}{p-2}$ and the maximum of $\phi _\theta$ on $[0,\frac{\pi }{2})$ is larger than $\beta _*$ and achieved at some $\bar{\theta }<\frac{\pi }{2}$ the exists $\theta _1<\bar{\theta }$ such that $\phi _\theta (\theta _1)=\beta _*$ and $\phi _{\theta \theta }(\theta _1)\ge 0$. In that case

$$\begin{aligned} 0\le A(\theta _1)=-{\frac{N-2 }{\Lambda _{\beta _*}-\beta _*} }\left( \beta _*(N-1)-\Lambda _{\beta _*}\right) \cot ^2\theta _1<0 \end{aligned}$$

which is again a contradictions.

Step 4: end of the proof Since $r^2=\beta ^2_*\omega ^2+\omega ^2_\theta$, $r_\theta =r(\phi _\theta -\beta _*)\tan \phi$, we have

$$\begin{aligned} rr_\theta =\left( \beta ^2_*\omega +\omega _{\theta \theta }\right) \omega _\theta = r(\phi _\theta -\beta _*)\tan \phi . \end{aligned}$$

Since $\omega _\theta < 0$ on $(0,\frac{\pi }{2})$, it follows from Step 3 that $\beta ^2_*\omega +\omega _{\theta \theta }\ge 0$ and thus

$$\begin{aligned} {\int _{0}^{\frac{\pi }{2}}}{\frac{\beta _*^2\omega +\omega _{\theta \theta }}{\beta ^2_*\omega ^2+\omega ^2_{\theta }} }\omega ^2_{\theta } \omega \cos \theta \sin ^{N-2}\theta d\theta >0, \end{aligned}$$

since the integrand cannot be identically 0. The conclusion follows from (6.9). $\square$

Remark

$\omega _{\theta }(\frac{\pi }{2})=-c^2<0$, it follows $\omega (\theta )=-\omega _\theta (\theta )\cot \theta +O(\frac{\pi }{2}-\theta )$ as $\theta \rightarrow \frac{\pi }{2}$, and from the eigenfunction Eq. (6.8)

$$\begin{aligned} {\frac{\beta _*^2\omega +\omega _{\theta \theta }}{\beta ^2_*\omega ^2+\omega ^2_{\theta }} }\omega ^2_{\theta }=(\beta _*^2\omega +\omega _{\theta \theta })(1+o(1)). \end{aligned}$$

Therefore

$$\begin{aligned} -(p-1)\omega _{\theta \theta }=(\beta _*\Lambda _{\beta _*}+(p-2)\beta _*^2+2-N)\omega (1+o(1))\quad \text {as }\theta \rightarrow \frac{\pi }{2} \end{aligned}$$

and since $\Delta '\omega :=\omega _{\theta \theta }+(N-2)\cot \theta \,\omega _{\theta }$

$$\begin{aligned} -\Delta '\omega =\frac{\beta _*(\beta _*(2p-3)+p-N)+(p-2)(N-2)}{p-1}\omega (1+o(1))\quad \text {as }\theta \rightarrow \frac{\pi }{2}. \end{aligned}$$

Because $\omega$ is $C^\infty$ we obtain finally

$$\begin{aligned} \left| \Delta '\omega \right| \le c\omega , \end{aligned}$$

(6.19)

for some $c>0$.