A priori estimates for elliptic equations with reaction terms involving the function and its gradient

Abstract

We study local and global properties of positive solutions of $-\Delta u=u^p+M\left| \nabla u\right| ^q$ in a domain $\Omega$ of $\mathbb {R}^N$, in the range $\min \{p,q\}>1$ and $M\in \mathbb {R}$. We prove a priori estimates and existence or non-existence of ground states for the same equation.

1 Introduction

This article is concerned with local and global properties of positive solutions of the following type of equations

$$\begin{aligned} -\Delta u=M'|u|^{p-1}u+M\left| \nabla u\right| ^q, \end{aligned}$$

(1.1)

in $\Omega {\setminus }\{0\}$ where $\Omega$ is an open subset of $\mathbb {R}^N$ containing 0, p and q are exponents larger than 1 and $M,M'$ are real parameters. If $M'\le 0$ the equation satisfies a comparison principle and a big part of the study can be carried via radial local supersolutions. This no longer the case when $M'>0$ which will be assumed in all the article, and by homothety (1.1) becomes

$$\begin{aligned} -\Delta u=|u|^{p-1}u+M\left| \nabla u\right| ^q. \end{aligned}$$

(1.2)

If $M=0$ (1.2) is called Lane–Emden equation

$$\begin{aligned} -\Delta u=|u|^{p-1}u. \end{aligned}$$

(1.3)

It turns out that it plays an important role in modelling meteorological or astrophysical phenomena [13, 15], this is the reason for which the first study, in the radial case, goes back to the end of nineteenth century and the beginning of the twentieth. A fairly complete presentation can be found in [18]. If $N\ge 3$, This equations exhibits two main critical exponents $p=\frac{N}{N-2}$ and $p=\frac{N+2}{N-2}$ which play a key role in the description of the set of positive solutions which can be summarized by the following overview:

• 1. If $1<p\le \frac{N}{N-2}$, there exists no positive solution if $\Omega$ is the complement of a compact set. Even in that case solution can be replaced by supersolution. This is easy to prove by studying the inequality satisfied by the spherical average of a solution of the equation.

• 2. If $1<p<\frac{N+2}{N-2}$, there exists no ground state, i.e. positive solution in $\mathbb {R}^N$. Furthermore any positive solution u in a ball $B_R=B_R(a)$ satisfies

  $$\begin{aligned} u(x)\le c(R-|x-a|)^{-\frac{2}{p-1}}, \end{aligned}$$

  (1.4)

  where $c=c(N,p)>0$, see [19].

• 3. If $p=\frac{N+2}{N-2}$ all the positive solutions in $\mathbb {R}^N$ are radial with respect to some point a and endow the following form

  $$\begin{aligned} u(x):=u_\lambda (x)=\frac{(N(N-2)\lambda )^{\frac{N-2}{4}}}{\left( \lambda +|x-a|^{2}\right) ^{\frac{N-2}{2}}}. \end{aligned}$$

  (1.5)

  All the positive solutions in $\mathbb {R}^N{\setminus }\{0\}$ are radial, see [12].

• 4. If $p>\frac{N+2}{N-2}$ there exist infinitely many positive ground states radial with respect to some points. They are obtained from one say v, radial for example with respect to 0 by the scaling transformation $T_k$ where $k>0$ with

  $$\begin{aligned} T_k[v](x)=k^{\frac{2}{p-1}}v(kx). \end{aligned}$$

  (1.6)

Indeed, the first significant non-radial results deals with the case $1<p\le \frac{N}{N-2}$. They are based upon the Brezis–Lions lemma [11] which yields an estimate of solutions in the Lorentz space $L^{\frac{N}{N-2},\infty }$, implying in turn the local integrability of $u^q$. Then a bootstrapping method as in [21] leads easily to some a priori estimate. Note that this subcritical case can be interpreted using the famous Serrin’s results on quasilinear equations [24]. The first breakthrough in the study of Lane–Emden equation came in the treatment of the case $1<p<\frac{N+2}{N-2}$; it is due to Gidas and Spruck [19]. Their analysis is based upon differentiating the equation and then obtaining sharp enough local integral estimates on the term $u^{q-1}$ making possible the utilization of Harnack inequality as in [24]. The treatment of the critical case $p=\frac{N+2}{N-2}$, due to Caffarelli, Gidas and Spruck [12], was made possible thanks to a completely new approach based upon a combination of moving plane analysis and geometric measure theory. As for the supercritical case, not much is known and the existence of radial ground states is a consequence of Pohozaev’s identity [22], using a shooting method.

The study of (1.2) when $M\ne 0$ presents some similarities with the one of Lane–Emden equation in the cases 1 and 2, except that the proof are much more involved. Actually the approach we develop in this article is much indebted to our recent paper [9] where we study local and global aspects of positive solutions of

$$\begin{aligned} -\Delta u=u^p\left| \nabla u\right| ^q, \end{aligned}$$

(1.7)

where $p\ge 0$, $0\le q<2$, mostly in the superlinear case $p+q-1>0$. Therein we prove the existence of a critical line of exponents

$$\begin{aligned} (\mathfrak {L}):=\{(p,q)\in \mathbb {R}_+\times [0,2): (N-2)p+(N-1)q=N\}. \end{aligned}$$

(1.8)

The subcritical range corresponds to the fact that (p, q) is below $(\mathfrak {L})$. In this region Serrin’s celebrated results [24] can be applied and we prove [9, Theorem A] that positive solutions of (1.7) in the punctured ball $B_2{\setminus }\{0\}$ satisfy, for some constant $c>0$ depending on the solution,

$$\begin{aligned} u(x)+\left| x\right| \left| \nabla u(x)\right| \le c\left| x\right| ^{2-N}\quad \text {for all }x\in B_{1}{\setminus }\{0\}. \end{aligned}$$

(1.9)

When (p, q) is above $(\mathfrak {L})$, i.e. in the supercritical range, we introduced two methods for obtaining a priori estimate of solutions: The pointwise Bernstein method and the integral Bernstein method. The first one is based upon the change of unknown $u=v^{-\beta }$, and then to show that $\left| \nabla v\right|$ satisfies an inequality of Keller–Osserman type. When (p, q) lies above $(\mathfrak {L})$ and verifies

• (i) either $1\le p<\frac{N+3}{N-1}$ and $p+q-1<\frac{4}{N-1}$,

• (ii) or $0\le p<1$ and $p+q-1<\frac{(p+1)^2}{p(N-1)}$,

we prove that any positive solution of (1.7) in a domain $\Omega \subset \mathbb {R}^N$ satisfies

$$\begin{aligned} \left| \nabla u^a(x)\right| \le c^*\left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-1-a\frac{2-q}{p+q-1}}\quad \text {for all }x\in \Omega , \end{aligned}$$

(1.10)

for some positive $c^*$ and a depending on N, p and q [9, Theorem B]. As a consequence we prove that any positive solution of (1.7) in $\mathbb {R}^N$ is constant. With the second method we combine the change of unknown $u=v^{-\beta }$ with integration and cut-off functions. We show the existence of a quadratic polynomial G in two variables such that for any $(p,q)\in \mathbb {R}_+\times [0,2)$ satisfying $G(p,q)<0$ any positive solution of (1.7) in $\mathbb {R}^N$ is constant [9, Theorem C]. The polynomial G is not simple but it is worth noting that if $0\le p<\frac{N+2}{N-2}$, there holds $G(p,0)<0$, which recovers Gidas and Spruck result [19].

For Eq. (1.2) we first observe that the equation is invariant under the scaling transformation (1.6) for any $k>0$ if and only if q is critical with respect to p, i.e.

$$\begin{aligned} q=\frac{2p}{p+1}. \end{aligned}$$

In general the transformation $T_k$ exchanges (1.2) with

$$\begin{aligned} -\Delta v=v^p+Mk^{\frac{2p-q(p+1)}{p-1}}|\nabla v|^q, \end{aligned}$$

(1.11)

hence if $q<\frac{2p}{p+1}$, the limit equation when $k\rightarrow 0$ is (1.3). We say that the exponent p is dominant. We can also consider the transformation

$$\begin{aligned} S_k[v](x)=k^{\frac{2-q}{q-1}}v(kx), \end{aligned}$$

(1.12)

when $q\ne 2$, which is the same as $T_k$ if $q=\frac{2p}{p+1}$, and more generally transforms (1.2) into

$$\begin{aligned} -\Delta v=k^{\frac{q-p(2-q)}{q-1}}v^p+M|\nabla v|^q. \end{aligned}$$

(1.13)

Hence if $q>\frac{2p}{p+1}$, the limit equation when $k\rightarrow 0$ is the Riccati equation

$$\begin{aligned} -\Delta v=M|\nabla v|^q. \end{aligned}$$

(1.14)

It is also important to notice that the value of the coefficient M (and not only its sign) plays a fundamental role, only if $q=\frac{2p}{p+1}$. If $q\ne \frac{2p}{p+1}$ the transformation

$$\begin{aligned} u(x)=av(y)\quad \text {with }\, a=\left| M\right| ^{-\frac{2}{(p+1)q-2p}}\text { and }\, y=a^{\frac{p-1}{2}}x \end{aligned}$$

(1.15)

allows to transform (1.2) into

$$\begin{aligned} -\Delta v=|v|^{p-1}v\pm |\nabla v|^q. \end{aligned}$$

(1.16)

Equation (1.2) has been essentially studied in the radial case when $M<0$ in connection with the parabolic equation

$$\begin{aligned} \partial _tu-\Delta u+M|\nabla u|^q=|u|^{p-1}u, \end{aligned}$$

(1.17)

see [14, 16, 17, 25, 27, 30, 31]. The studies mainly deal with the case $q\ne \frac{2p}{p+1}$, although not complete when $q>\frac{2p}{p+1}$. When $q=\frac{2p}{p+1}$ the existence of a ground state is proved in dimension 1. Some partial results that we will improve, already exist in higher dimension. The case $M>0$ attracted less attention.

In the nonradial case, any nonnegative nontrivial solution is positive since $p,q>1$. We first observe, using a standard averaging method applied to positive supersolutions of (1.3), that if $M\ge 0$, $1<p\le \frac{N}{N-2}$ when $N\ge 3$, any $p>1$ if $N=1,2$, then for any $q>0$ there exists no positive solution in an exterior domain. When $0<q<\frac{2p}{p+1}$ the equation endows some character of the pure Emden–Fowler equation (1.3) by the transformation $T_k$. In [23] it is proved that if $0<q<\frac{2p}{p+1}$, $1<p<\frac{N+2}{N-2}$ and $M\in \mathbb {R}$, any positive solution of (1.3) in an open domain satisfies

$$\begin{aligned} u(x)+\left| \nabla u(x)\right| ^{\frac{2}{p+1}}\le c_{N,p,q,M}\left( 1+\left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{2}{p-1}}\right) \quad \text {for all }x\in \Omega . \end{aligned}$$

(1.18)

Note that this does not imply the non-existence of ground state. In [1] Alarcón, García-Melián and Quass study the equation

$$\begin{aligned} -\Delta u=|\nabla u|^q+f(u), \end{aligned}$$

(1.19)

in an exterior domain of $\mathbb {R}^N$ emphasizing the fact that positive solutions are super harmonic functions. They prove that if $1<q\le \frac{N}{N-1}$ and if f is positive on $(0,\infty )$ and satisfies

$$\begin{aligned} \displaystyle \limsup _{s\rightarrow 0}s^{-p}f(s)>0, \end{aligned}$$

(1.20)

for some $p> \frac{N}{N-2}$, then (1.19) admits no positive supersolution. The same authors also study in [2] existence and non-existence of positive solutions of (1.19) in a bounded domain with Dirichlet condition.

The techniques we developed in this paper are based upon a delicate extension of the ones already introduced in [9]. Our first nonradial result dealing with the case $q>\frac{2p}{p+1}$ is the following:

Theorem A

Let $N\ge 1$, $p>1$ and $q>\frac{2p}{p+1}$. Then for any $M>0$, any solution of (1.2) in a domain $\Omega \subset \mathbb {R}^N$ satisfies

$$\begin{aligned} \left| \nabla u(x)\right| \le c_{N,p,q}\left( M^{-\frac{p+1}{(p+1)q-2p}}+\left( M\mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{1}{q-1}}\right) \quad \text {for all }x\in \Omega .\quad \end{aligned}$$

(1.21)

As a consequence, any ground state has at most a linear growth at infinity:

$$\begin{aligned} \left| \nabla u(x)\right| \le c_{N,p,q}M^{-\frac{p+1}{(p+1)q-2p}}\qquad \text {for all }x\in \mathbb {R}^N. \end{aligned}$$

(1.22)

Our proof relies on a direct Bernstein method combined with Keller–Osserman’s estimate applied to $\left| \nabla u\right| ^2$. It is important to notice that the result holds for any $p>1$, showing that, in some sense, the presence of the gradient term has a regularizing effect. In the case $q<\frac{2p}{p+1}$ we prove a non-existence result

Theorem A $'$ Let $N\ge 1$, $p>1$, $1<q<\frac{2p}{p+1}$ and $M>0$. Then there exists a constant $c_{N,p,q}>0$ such that there is no positive solution of (1.2) in $\mathbb {R}^N$ satisfying

$$\begin{aligned} u(x)\le c_{N,p,q}M^{\frac{2}{2p-(p+1)q}}\qquad \text {for all }x\in \mathbb {R}^N. \end{aligned}$$

(1.23)

When q is critical with respect to p the situation is more delicate since the value of M plays a fundamental role. Our first statement is a particular case of a more general result in [1], but with a simpler proof which allows us to introduce techniques that we use later on.

Theorem B

Let $N\ge 2$, $p>1$ if $N=2$ or $1<p\le \frac{N}{N-2}$ if $N=3$, $q=\frac{2p}{p+1}$ and $M>-\mu ^*$ where

$$\begin{aligned} \mu ^*:=\mu ^*(N)=(p+1)\left( {\displaystyle \frac{N-(N-2)p}{2p} }\right) ^{\frac{p}{p+1}}. \end{aligned}$$

(1.24)

Then there exists no nontrivial nonnegative supersolution of (1.2) in an exterior domain.

In this range of values of p this result is optimal since for $M\le -\mu ^*$ there exists positive singular solutions. The constant $\mu ^*$ will play an important role in the description developed in [10] of radial solutions of (1.2). Using a variant of the method used in the proof of Theorem B we obtain results of existence and nonexistence of large solutions.

Theorem B$'$ Let $N\ge 1$, $p>1$ and $q=\frac{2p}{p+1}$.

• 1. If $\Omega$ is a domain with a compact boundary satisfying the Wiener criterion and $M\ge -\mu ^*(2)$ there exists no positive supersolution of (1.2) in $\Omega$ satisfying

  $$\begin{aligned} \displaystyle \lim _{\mathrm{dist}\,(x,\partial \Omega )\rightarrow 0}u(x)=\infty . \end{aligned}$$

  (1.25)

• 2. If G is a bounded convex domain, $\Omega =\overline{G}^c$ and $M< -\mu ^*(1)$ there exists a positive solution of (1.2) in $\Omega$ satisfying (1.25).

We show in [10] that the inequality $M< -\mu ^*(1)$ is the necessary and sufficient condition for the existence of a radial large solution in the exterior of a ball.

Concerning ground states, we prove their nonexistence for any $p>1$ provided $M>0$ is large enough: indeed

Theorem C

Let $\Omega \subset \mathbb {R}^N$, $N\ge 1$, be a domain, $p>1$, $q=\frac{2p}{p+1}$. For any

$$\begin{aligned} M>M_\dag :=\left( \frac{p-1}{p+1}\right) ^{\frac{p-1}{p+1}} \left( \frac{N(p+1)^2}{4p}\right) ^{\frac{p}{p+1}}, \end{aligned}$$

(1.26)

and any $\nu >0$ such that $(1-\nu )M>M_\dag$, there exists a positive constant $c_{N,p,\nu }$ such that any solution u in $\Omega$ satisfies

$$\begin{aligned} \left| \nabla u(x)\right| \le c_{N,p,\nu }\left( (1-\nu )M-M_\dag \right) ^{-\frac{p+1}{p-1}}\left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{p+1}{p-1}}\quad \text {for all }x\in \Omega . \end{aligned}$$

(1.27)

Consequently there exists no nontrivial solution of (1.2) in $\mathbb {R}^N$.

The next result, based upon an elaborate Bernstein method, complements Theorem C under a less restrictive assumption on M but a more restrictive assumption on p.

Theorem D

Let $1<p<\frac{N+3}{N-1}$, $N\ge 2$, $1<q<\frac{N+2}{N}$ and $\Omega \subset \mathbb {R}^N$ be a domain. Then there exist $a>0$ and $c_{N,p,q}>0$ such that for any $M>0$, any positive solution u in $\Omega$ satisfies

$$\begin{aligned} \left| \nabla u^{a}(x)\right| \le c_{N,p,q}\left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{2a}{p-1}-1}\quad \text {for all }x\in \Omega . \end{aligned}$$

(1.28)

Hence there exists no nontrivial nonnegative solution of (1.2) in $\mathbb {R}^N$.

It is remarkable that the constants a and $c_{N,p,q}$ do not depend on $M>0$, a fact which is clear when $q\ne \frac{2p}{p+1}$ by using the transformation $T_k$, but much more delicate to highlight when $q=\frac{2p}{p+1}$ since (1.2) is invariant. When $\left| M\right|$ is small, we use an integral method to obtain the following result which contains, as a particular case, the estimates in [19] and [10]. The key point of this method is to prove that the solutions in a punctured domain satisfy a local Harnack inequality.

Theorem E

Let $N\ge 3$, $1<p<\frac{N+2}{N-2}$, $q=\frac{2p}{p+1}$. Then there exists $\epsilon _0>0$ depending on N and p such that for any M satisfying $\left| M\right| \le \epsilon _0$, any positive solution u in $B_R{\setminus }\{0\}$ satisfies

$$\begin{aligned} u(x)\le c_{N,p}\left| x\right| ^{-\frac{2}{p-1}}\quad \text {for all }x\in B_{\frac{R}{2}}{\setminus }\{0\}. \end{aligned}$$

(1.29)

As a consequence there exists no positive solution of (1.2) in $\mathbb {R}^N$, and any positive solution u in a domain $\Omega$ satisfies

$$\begin{aligned} u(x)+\left| \nabla u(x) \right| ^{\frac{2}{p+1}}\le c'_{N,p}\left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{2}{p-1}}\quad \text {for all }x\in \Omega . \end{aligned}$$

(1.30)

Note that under the assumptions of Theorem E, there exist ground states for $\left| M\right|$ large enough when $1<p<\frac{N}{N-2}$, or any $p>1$ if $N=1,2$.

If u is a radial solutions of (1.2) in $\mathbb {R}^N$ it satisfies

$$\begin{aligned} -u''-{\displaystyle \frac{N-1}{r} }u'=\left| u\right| ^{p-1}u+M\left| u'\right| ^{q}, \end{aligned}$$

(1.31)

on $(0,\infty )$. Using several type of Lyapounov type functions introduced by Leighton [20] and Anderson and Leighton [3], we prove some results dealing with the case $M>0$ which complement the ones of [25] relative to the case $M<0$.

Theorem F

• 1. Let $p>1$ and $q>\frac{2p}{p+1}$. Then there exists no radial ground state u satisfying $u(0)=1$ when $M>0$ is too large.

• 2. Let $1<p<\frac{N+2}{N-2}$. If $1<q\le p$ there exists no radial ground state for any $M>0$. If $q>p$ there exists no radial ground state for $M>0$ small enough.

• 3. Let $N\ge 3$, $p>\frac{N+2}{N-2}$ and $q\ge \frac{2p}{p+1}$. Then there exist radial ground states for $M>0$ small enough.

We end the article in proving the existence of non-radial positive singular solutions of (1.2) in $\mathbb {R}^N{\setminus }\{0\}$ in the case $q=\frac{2p}{p+1}$ obtained by bifurcation from radial explicit positive singular solutions. Our result shows that the situation is very contrasted according $M>0$ where a bifurcation from $(M,X_{M})$ occurs only if $p\ge \frac{N+1}{N-3}$ and $M\ge 0$ and $M<0$ where there exists a countable set of bifurcations from $(M_k,X_{M_k})$, $k\ge 1$, when $1<p<\frac{N+1}{N-3}$.

In a subsequent article [10] we present a fairly complete description of the positive radial solutions of (1.2) in $\mathbb {R}^N{\setminus }\{0\}$ in the scaling invariant case $q=\frac{2p}{p+1}$.

2 The direct Bernstein method

We begin with a simple property in the case $M\ge 0$ which is a consequence of the fact that the positive solutions of (1.2) are superharmonic.

Proposition 2.1

• 1. There exists no positive solution of (1.2) in $\mathbb {R}^N{\setminus }\overline{B}_R$, $R\ge 0$ if one of the two conditions is satisfied:

  • (i) $M\ge 0$, $q\ge 0$ and either $N=1,2$ and $p>1$ or $N\ge 3$ and $1<p\le \frac{N}{N-2}$.

  • (ii) $M> 0$, $N\ge 3$, $p\ge 1$ and $1<q\le \frac{N}{N-1}$.

• 2. If $N\ge 3$, $q\ge 1$, $p>\frac{N}{N-2}$ and $u(x)=u(r,\sigma )$ is a positive solution of (1.2) in $\mathbb {R}^N{\setminus }\overline{B}_R$, $R\ge 0$. Then there exists $\rho \ge R$ such that

  $$\begin{aligned} {\displaystyle \frac{1}{N\omega _N} }{\displaystyle \int _{S^{N-1}}^{}}u(r,\sigma )dS:=\overline{u}(r)\le c_0r^{-\frac{2}{p-1}}\qquad \text {for all }\;r>\rho , \end{aligned}$$

  (2.1)

  with $c_0:=\left( \frac{2N}{p-1}\right) ^{\frac{1}{p-1}}$ and

  $$\begin{aligned} \left| {\displaystyle \frac{1}{N\omega _N} }{\displaystyle \int _{S^{N-1}}^{}}u_r(r,\sigma )dS\right| :=\left| \overline{u}_r(r)\right| \le (N-2)c_0r^{-\frac{p+1}{p-1}}\quad \text {for all }\;r>\rho .\ \ \ \end{aligned}$$

  (2.2)

• 3. If $M>0$, $p\ge 0$, and $q>\frac{N}{N-1}$ there holds for

  $$\begin{aligned} \left| \overline{u}_r(r)\right| \le \left( {\displaystyle \frac{(q-1)(N-1)-1}{(q-1)M} }\right) ^{\frac{1}{q-1}}r^{-\frac{1}{q-1}}\qquad \text {for all }\;r>\rho , \end{aligned}$$

  (2.3)

  and

  $$\begin{aligned} \overline{u}(r)\le \Bigl ({\displaystyle \frac{q-1}{2-q} }\Bigr )\left( {\displaystyle \frac{(q-1)(N-1)-1}{(q-1)M} }\right) ^{\frac{1}{q-1}}r^{\frac{q-2}{q-1}}\qquad \text {for all }\;r>\rho , \end{aligned}$$

  (2.4)

  Furthermore, if $R=0$, inequalities (2.1), (2.2) and (2.3) hold with $\rho =0$.

Proof

Assertion 1-(i) is not difficult to obtain by integrating the inequality satisfied by the spherical average of the solution and using Jensen’s inequality. For the sake of completeness, we give a simple proof although the result is actually valid for much more general equations (see e.g. [5] and references therein). In this statement we denote by $(r,\sigma )\in \mathbb {R}_+\times S^{N-1}$ the spherical coordinates in $\mathbb {R}^N$, by $\omega _N$ the volume of the unit N-ball and thus $N\omega _N$ is the (N-1)-volume of the unit sphere $S^{N-1}$. Writing (1.2) in spherical coordinates and using Jensen formula, we get

$$\begin{aligned} -r^{1-N}\left( r^{N-1}\overline{u}_r\right) _r\ge \overline{u}^p+M\left| \overline{u}_r\right| ^q. \end{aligned}$$

(2.5)

It implies that $r\mapsto w(r):= -r^{N-1}\overline{u}_r$ is increasing on $(R,\infty )$, thus it admits a limit $\ell \in (-\infty ,\infty ]$. If $\ell \le 0$, then $\overline{u}_r(r)> 0$ on $(R,\infty )$. Hence $\overline{u}(r)\ge \overline{u}(\rho ):=c>0$ for $r\ge \rho >R$. then

$$\begin{aligned} \left( r^{N-1}\overline{u}_r\right) _r\le -c^pr^{N-1}\Longrightarrow \overline{u}_r(r)\le {\displaystyle \frac{\rho ^{N-1}}{r^{N-1}} }\overline{u}_r(\rho ) -{\displaystyle \frac{c^p}{N} }\left( r-{\displaystyle \frac{\rho ^{N}}{r^{N-1}} }\right) , \end{aligned}$$

which implies $\overline{u}_r(r)\rightarrow -\infty$, thus $\overline{u}(r)\rightarrow -\infty$ as $r\rightarrow -\infty$, contradiction. Therefore $\ell \in (0,\infty ]$ and either $\overline{u}_r(r)<0$ on $(R,\infty )$ or there exists $r_{\ell }>R$ such that $\overline{u}_r(r_\ell )=0$, $\overline{u}$ is increasing on $(R,r_\ell ,)$ and decreasing on $(r_\ell ,\infty )$. If $\overline{u}_r(r)<0$ on $(R,\infty )$, then we have for $r> 2R$

$$\begin{aligned} -r^{N-1}\overline{u}_r(r)\ge & {} {\displaystyle \int _{\frac{r}{2}}^{r}}t^{N-1}\overline{u}^p(t)dt\ge {\displaystyle \frac{r^{N}\overline{u}^p(r)}{2N} } \Longrightarrow \left( \overline{u}^{1-p}\right) _r\\\ge & {} {\displaystyle \frac{(p-1)r}{2N} }\Longrightarrow \overline{u}(r)\le \left( {\displaystyle \frac{2N}{(p-1)r^2} }\right) ^{\frac{1}{p-1}}, \end{aligned}$$

which yields (2.1). If we are in the second case with $r_\ell >R$, we apply the same inequality with $r> 2r_\ell$ and again (2.1) for $r> 2r_\ell$. Since $\overline{u}$ is superharmonic, the function $v(s)=\overline{u}(r)$ with $s=r^{2-N}$ is concave on $(0,R^{2-N})$ and it tends to 0 when $s\rightarrow 0$. Thus

$$\begin{aligned} v_s(s)\le \frac{v}{s}\Longrightarrow \left| \overline{u}_r(r)\right| \le (N-2){\displaystyle \frac{\overline{u}(r)}{r} }\le (N-2)c_0r^{-\frac{p+1}{p-1}}. \end{aligned}$$

This implies (2.1) and (2.2). Note that the case $r_\ell >R$ cannot happen if $R=0$, so in any case, if $R=0$ then $\rho =0$.

If $M>0$, we have with $w(r)=-r^{N-1}\overline{u}_r$

$$\begin{aligned} w_r\ge Mr^{(1-q)(N-1)}\left| w\right| ^q. \end{aligned}$$

We have seen that $w(r)>0$ at infinity with limit $\ell \in (0,\infty ]$, hence, on the maximal interval containing $\infty$ where $w>0$, we have $(w^{1-q})_r\le (1-q)Mr^{(N-1)(1-q)}$. We have for $r>s>R$

$$\begin{aligned} w^{1-q}(r)-w^{1-q}(s)\le M\ln \left( \frac{r}{s}\right) , \end{aligned}$$

if $q=\frac{N}{N-1}$ and

$$\begin{aligned} w^{1-q}(r)-w^{1-q}(s)\le \frac{M(q-1)}{(q-1)(N-1)-1}\left( r^{1-(q-1)(N-1)}-s^{1-(q-1)(N-1)}\right) \end{aligned}$$

if $q<\frac{N}{N-1}$, and both expressions which tend to $-\infty$ when $r\rightarrow \infty$, a contradiction. This proves 1-(ii). If $q>\frac{N}{N-1}$, the above expression yields, when $r\rightarrow \infty$,

$$\begin{aligned} \ell ^{1-q}-w^{1-q}(s)\le -{\displaystyle \frac{(q-1)M}{(q-1)(N-1)-1} }s^{1-(q-1)(N-1)}. \end{aligned}$$

This implies

$$\begin{aligned} w(s)\le \left( {\displaystyle \frac{(q-1)(N-1)-1}{(q-1)M} }\right) ^{\frac{1}{q-1}}s^{N-1-\frac{1}{q-1}}, \end{aligned}$$

and (2.3).$\square$

Remark

The previous is a particular case of a much more general one dealing with quasilinear operators proved in [5, Theorem 3.1].

2.1 Proof of Theorems A, A$'$ and C

The function u is at least $C^{3+\alpha }$ for some $\alpha \in (0,1)$ since $p,q>1$. Hence $z=\left| \nabla u\right| ^2$ is $C^{2+\alpha }$. Since there holds by Bochner’s identity and Schwarz’s inequality

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{1}{N} }(\Delta u)^2+\langle \nabla \Delta u,\nabla u\rangle \le 0, \end{aligned}$$

(2.6)

we obtain from (1.2),

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{|u|^{2p}}{N} }+{\displaystyle \frac{2M}{N} }|u|^{p-1}uz^{\frac{q}{2}}+{\displaystyle \frac{M^2}{N} }z^q-p|u|^{p-1}z -{\displaystyle \frac{Mq}{2} }z^{\frac{q}{2}-1}\langle \nabla z,\nabla u\rangle \le 0. \end{aligned}$$

Since for $\delta >0$,

$$\begin{aligned} z^{\frac{q}{2}-1}\left| \langle \nabla z,\nabla u\rangle \right| \le \left| z^{-\frac{1}{2}}\nabla z\right| z^{\frac{q-1}{2}}\left| \nabla u\right| = \left| z^{-\frac{1}{2}}\nabla z\right| z^{\frac{q}{2}}\le \delta z^q+{\displaystyle \frac{1}{4\delta } }{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }, \end{aligned}$$

we obtain for any $\nu \in (0,1)$, provided $\delta$ is small enough,

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{|u|^{2p}}{N} }+{\displaystyle \frac{2M}{N} }|u|^{p-1}uz^{\frac{q}{2}}+{\displaystyle \frac{M^2(1-\nu )^2}{N} }z^q-p|u|^{p-1}z \le c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} },\quad \end{aligned}$$

(2.7)

where $c_1=c_1(M,N,\nu )>0$.

2.1.1 Proof of Theorem A

We recall the following technical result proved in [9, Lemma 2.2] which will be used several times in the course of this article.

Lemma 2.2

Let $S>1$, $R>0$ and v be continuous and nonnegative in $\overline{B}_R$ and $C^1$ on the set ${\mathcal {U}}_+=\{x\in B_R:v(x)>0\}$. If v satisfies, for some real number a,

$$\begin{aligned} -\Delta v+v^S\le a{\displaystyle \frac{|\nabla v|^2}{v} } \end{aligned}$$

(2.8)

on each connected component of ${\mathcal {U}}_+$, then

$$\begin{aligned} v(0)\le c_{N,S,a}R^{-\frac{2}{S-1}}. \end{aligned}$$

(2.9)

Abridged proof

Assuming $a>0$, we set $W=v^\alpha$ for $0<\alpha \le \frac{1}{a+1}$, this transforms (2.8) into

$$\begin{aligned} -\Delta W+{\displaystyle \frac{1}{\alpha } }W^{\alpha (S-1)+1}\le 0, \end{aligned}$$

(2.10)

and then we apply Keller–Osserman inequality. $\square$

Proof of Theorem A

Suppose $\frac{2p}{p+1}<q$. We set $r=\frac{2p}{p-1}$, $r'=\frac{r}{r-1}$, then, for any $\epsilon >0$

$$\begin{aligned} p|u|^{p-1}z\le {\displaystyle \frac{\epsilon ^r|u|^{(p-1)r}}{r} }+{\displaystyle \frac{z^{r'}}{\epsilon ^{r'}r'} }=(p-1){\displaystyle \frac{\epsilon ^r|u|^{2p}}{2} } +(p+1){\displaystyle \frac{z^{\frac{2p}{p+1}}}{2\epsilon ^{r'}} }. \end{aligned}$$

We fix $\eta \in (0,1)$ and $\epsilon$ so that $\epsilon ^r=\frac{2(1-\eta )}{N(p-1)}$ and get

$$\begin{aligned} p|u|^{p-1}z\le (1-\eta )\frac{|u|^{2p}}{N}+c_2z^{\frac{2p}{p+1}}, \end{aligned}$$

where $c_2=\frac{p+1}{2}\left( \frac{N(p-1)}{2(1-\eta )}\right) ^{\frac{p+1}{p-1}}$. We perform the change of scale (1.6) in order to reduce (1.2) to the case $M=1$ by setting $u(x)=\alpha ^{\frac{2}{p-1}}v(\alpha x)$ with $\alpha =M^{-\frac{p-1}{(p+1)q-2p}}$. Then the equation for $z=\left| \nabla v\right| ^2$ is considered in $\Omega _\alpha =\alpha \Omega$. Choosing now $\eta =\frac{1}{2}$ we obtain

$$\begin{aligned} c_2z^{\frac{2p}{p+1}}\le {\displaystyle \frac{1}{4N} }z^q+c_3, \end{aligned}$$

where $c_3=c_3(N,p,q)>0$, hence

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{v^{2p}}{2N} }+{\displaystyle \frac{1}{4N} }z^q\le c_3+c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }. \end{aligned}$$

Put $\tilde{z}=\left( z-\left( 4Nc_3\right) ^{\frac{1}{q}}\right) _+$, then

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta \tilde{z}+{\displaystyle \frac{1}{4N} }\tilde{z}^q\le c_1{\displaystyle \frac{\left| \nabla \tilde{z}\right| ^2}{\tilde{z}} }, \end{aligned}$$

hence, from Lemma 2.2, we derive

$$\begin{aligned} \tilde{z}(y)\le c_4\left( \mathrm{dist}\,(y,\partial \Omega _\alpha )\right) ^{\frac{2}{q-1}} \end{aligned}$$

where $c_4=c_4(N,q,c_1)>0$ which implies

$$\begin{aligned} \left| \nabla v(y)\right| \le c'_4\left( 1+\left( \mathrm{dist}\,(y,\partial \Omega _\alpha )\right) ^{-\frac{1}{q-1}}\right) \qquad \forall \, y\in \Omega _\alpha . \end{aligned}$$

(2.11)

Then (1.21) and (1.22) follow.

Assume now that there exists a ground state u. Fix $y\in \mathbb {R}^N$ and consider $\{y_n\}\subset \mathbb {R}^N$ such that $|y_n|=2n>|y|$. We apply (2.11) with $\Omega _\alpha =B_n(y_n)$. Then

$$\begin{aligned} \left| \nabla v(y)\right| \le c'_4\left( 1+\left| 2n-|y|\right| ^{-\frac{1}{q-1}}\right) , \end{aligned}$$

and letting $n\rightarrow \infty$ we infer

$$\begin{aligned} \left| \nabla v(y)\right| \le c'_4\qquad \forall \, y\in \mathbb {R}^N. \end{aligned}$$

(2.12)

Hence, by the definition of v and y we see that

$$\begin{aligned} |\nabla u(x)|\le c'_4M^{-\frac{p+1}{(p+1)q-2p}}\qquad \forall \, x\in \mathbb {R}^N \end{aligned}$$

which is exactly (1.22). $\square$

2.1.2 Proof of Theorem A$'$

Suppose $1<q<\frac{2p}{p+1}$. By scaling we reduce to the case $M=1$ and we replace u by v defined by (1.6) as in the proof of Theorem A with $\alpha =M^{\frac{p-1}{2p-(p+1)q}}$. From (2.7) with $\nu =\frac{1}{4}$ the function $z=\left| \nabla v\right| ^2$ satisfies

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{v^{2p}}{N} }+{\displaystyle \frac{1}{2N} }z^q-pv^{p-1}z \le c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }. \end{aligned}$$

(2.13)

By Hölder’s inequality,

$$\begin{aligned} pv^{p-1}z\le {\displaystyle \frac{1}{4N} }z^q+p(4Np)^{q'-1}v^{(p-1)q'}. \end{aligned}$$

Since $(p-1)q'=2p+\frac{2p-(p+1)q}{q-1}$ we derive

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{v^{2p}}{N} }\left( 1-4^{q'-1}p^{q'}N^{q'}v^{\frac{2p-(p+1)q}{q-1}}\right) +{\displaystyle \frac{1}{4N} }z^q\le c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }. \end{aligned}$$

If $\max v\le c_{N,p,q}:=(4^{q'-1}p^{q'}N^{q'})^{-\frac{q-1}{2p-(p+1)q}}$, we obtain

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{1}{4N} }z^q\le c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }, \end{aligned}$$

which implies that $z=0$ by Lemma 2.2, hence v is constant and thus $v=0$ from the equation. $\square$

Remark

If u is a positive ground state of (1.2) radial with respect to 0, it satisfies $u_r(0)=0$ and it is a decreasing function of r. The previous theorem asserts that it must satisfy

$$\begin{aligned} u(0)>c_{N,p,q} M^{\frac{2}{2p-(p+1)q}}. \end{aligned}$$

(2.14)

2.1.3 Proof of Theorem C

Suppose $\frac{2p}{p+1}=q$. For $A>0$ we consider the expression

$$\begin{aligned}&\left( u^p+A\left| \nabla u\right| ^{q}\right) ^2-Npu^{p-1}\left| \nabla u\right| ^2\\&\quad =\left( u^p+A\left| \nabla u\right| ^{q}-\sqrt{Np\,}u^{\frac{p-1}{2}}\left| \nabla u\right| \right) \left( u^p+A\left| \nabla u\right| ^{q}+\sqrt{Np\,}u^{\frac{p-1}{2}}\left| \nabla u\right| \right) . \end{aligned}$$

Now the function $Z\mapsto \Phi _A(Z)=u^{p}+AZ^{q}-\sqrt{Np\,}\,u^{\frac{p-1}{2}}Z$ achieves its minimum at $Z_0=\left( \frac{\sqrt{Np}}{qA }\right) ^{\frac{p+1}{p-1}}u^{\frac{p+1}{2}}$ and

$$\begin{aligned} \Phi _A(Z_0)=\left[ 1-{\displaystyle \frac{p-1}{p+1} }\left( {\displaystyle \frac{N(p+1)^2}{4p} }\right) ^{\frac{p}{p-1}}A^{-\frac{p+1}{p-1}}\right] u^{p}. \end{aligned}$$

Thus setting

$$\begin{aligned} M_\dag =\left( \frac{p-1}{p+1}\right) ^{\frac{p-1}{p+1}}\left( \frac{N(p+1)^2}{4p} \right) ^{\frac{p}{p+1}}, \end{aligned}$$

(2.15)

we obtain that if $A\ge M_\dag$, then $\Phi _A(Z)\ge 0$ for all Z. Put $M_\nu =(1-\nu ) M$ for $\nu \in (0,1)$ such that $M_\dag <M_\nu$, we derive from (2.7)

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{(u^{p}+M_\dag z^{\frac{q}{2}})^2}{N} }-pu^{p-1}z+{\displaystyle \frac{M_\nu ^2-M_\dag ^2}{N} }z^q \le c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }, \end{aligned}$$

(2.16)

which yields

$$\begin{aligned} -{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{M_\nu ^2-M_\dag ^2}{N} }z^q \le c_1{\displaystyle \frac{\left| \nabla z\right| ^2}{z} }. \end{aligned}$$

Using again Lemma 2.2 we obtain

$$\begin{aligned} \left| \nabla u(x)\right| \le c'_1\left( (1-\nu )M-M_\dag \right) ^{-\frac{1}{q-1}}\left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{1}{q-1}}, \end{aligned}$$

(2.17)

which is equivalent to (1.27). $\square$

2.2 Proof of Theorems B and B$'$

2.2.1 Proof of Theorem B

Since the result is known when $M\ge 0$ from Proposition 2.1, we can assume that $M=-m<0$ and $N=1,2$ or $N\ge 3$ with $p<\frac{N}{N-2}$, u is a nonnegative supersolution of (1.2) in $\overline{B}^c_R$ and we set $u=v^b$ with $b>1$. Then

$$\begin{aligned} -\Delta v\ge (b-1)\frac{\left| \nabla v\right| ^2}{v}+\frac{1}{b}v^{1+b(p-1)}-mb^{q-1}v^{(b-1)(q-1)}\left| \nabla v\right| ^q. \end{aligned}$$

(2.18)

Here again $q=\frac{2p}{p+1}$, setting $z=\left| \nabla v\right| ^2$ we obtain

$$\begin{aligned} -\Delta v\ge \frac{\Phi (z)}{bv} \end{aligned}$$

where

$$\begin{aligned} \Phi (z)=b(b-1)z-mb^{\frac{2p}{p+1}}v^{\frac{2+b(p-1)}{p+1}}z^{\frac{p}{p+1}} +v^{2+b(p-1)}. \end{aligned}$$

Thus $\Phi$ achieves it minimum for

$$\begin{aligned} z_0=\left( \frac{mpb^{q-1}}{(b-1)(p+1)}\right) ^{p+1}b^{p-1}v^{2+b(p-1)} \end{aligned}$$

and

$$\begin{aligned} \Phi (z_0)=v^{2+b(p-1)}\left( 1-\frac{p^p}{(p+1)^{p+1}}\left( \frac{b}{b-1}\right) ^pm^{p+1}\right) . \end{aligned}$$

(2.19)

In order to ensure the optimal choice, when $N\ge 3$ we take $1+b(p-1)=\frac{N}{N-2}$, hence $b=\frac{2}{(N-2)(p-1)}$ which is larger than 1 because $p<\frac{N}{N-2}$. Finally

$$\begin{aligned} \Phi (z_0)=v^{\frac{N}{N-2}+1}\left( 1-\frac{1}{(p+1)^{p+1}}\left( \frac{2p}{N-p(N-2)}\right) ^pm^{p+1}\right) . \end{aligned}$$

Hence, if

$$\begin{aligned} m< (p+1)\left( \frac{N-p(N-2)}{2p}\right) ^{\frac{p}{p+1}}=\mu ^*(N), \end{aligned}$$

(2.20)

we have for some $\delta >0$,

$$\begin{aligned} -\Delta v\ge \delta v^{\frac{N}{N-2}}, \end{aligned}$$

(2.21)

and by Proposition 2.1 that is no positive solution in an exterior domain of $\mathbb {R}^N$.

If $N=2$ for a given $b>1$ we have from (2.19) that if

$$\begin{aligned} m<(p+1)\left( \frac{b-1}{bp}\right) ^{\frac{p}{p+1}}, \end{aligned}$$

then, for some $\delta >0$,

$$\begin{aligned} -\Delta v\ge \delta v^{1+b(p-1)}. \end{aligned}$$

(2.22)

The result follows from Proposition 2.1 by choosing b large enough. $\square$

2.2.2 Proof of Theorem B$'$

1. We assume that such a supersolution u exists and we denote $u=e^v$, then

$$\begin{aligned} -\Delta v\ge F(\left| \nabla v\right| ^2), \end{aligned}$$

(2.23)

where

$$\begin{aligned} F(X)=X+e^{(p-1)v}+Me^{\frac{p-1}{p+1}v}X^{\frac{p}{p+1}}. \end{aligned}$$

Clearly, if $M\ge 0$, then $F(X)\ge 0$ for any $X\ge 0$. Next we assume $M<0$, then

$$\begin{aligned} F(X)\ge F(X_0)=e^{(p-1)v}\left( 1-p^p\left( \frac{\left| M\right| }{p+1}\right) ^{p+1}\right) =e^{(p-1)v}\left( 1-\left( \frac{\left| M\right| }{\mu ^*(2)}\right) ^{p+1}\right) . \end{aligned}$$

Hence, if $\left| M\right| \le \mu ^*(2)$, v is a positive superharmonic function in $\Omega$ which tends to infinity on the boundary. Such a function is larger than the harmonic function with boundary value $k>0$ for any k (and taking the value $\min _{\left| x\right| =R}v(x)$ for R large enough if $\Omega$ is an exterior domain). Letting $k\rightarrow \infty$ we derive a contradiction.

2. Let $R>0$ such that $\Omega ^c\subset B_R$ and let w be the solution of

$$\begin{aligned} \begin{array}{ll} -\Delta w-ae^{(p-1)w}=0\qquad &{}\text {in }B_R\cap \Omega \\ \displaystyle \lim _{\mathrm{dist}\,(x,\partial B_R)\rightarrow 0}w(x)=-\infty \\ \displaystyle \lim _{\mathrm{dist}\,(x,\partial \Omega )\rightarrow 0}w(x)=\infty , \end{array} \end{aligned}$$

(2.24)

with $a=1-\left( \frac{\left| M\right| }{\mu ^*(2)}\right) ^{p+1}<0$, obtained by approximations. By the argument used in 1,

$$\begin{aligned} ae^{(p-1)w}\le \left| \nabla w\right| ^2+e^{(p-1)w}-\left| M\right| e^{\frac{p-1}{p+1}w}\left| \nabla w\right| ^{\frac{2p}{p+1}}, \end{aligned}$$

hence

$$\begin{aligned} -\Delta w\le \left| \nabla w\right| ^2+e^{(p-1)w}-\left| M\right| e^{\frac{p-1}{p+1}w}\left| \nabla w\right| ^{\frac{2p}{p+1}}. \end{aligned}$$

Therefore $v=e^{w}$ is nonnegative and satisfies

$$\begin{aligned} \begin{array}{rl} -\Delta v-v^p+\left| M\right| \left| \nabla v\right| ^{\frac{2p}{p+1}}\le 0&{}\qquad \text {in }B_R\cap \Omega \\ v=0&{}\qquad \text {on }\partial B_R\\ \displaystyle \lim _{\mathrm{dist}\,(x,\partial \Omega )\rightarrow 0}v(x)=\infty .&{} \end{array} \end{aligned}$$

(2.25)

Next we extend v by zero in $B_R^c$ and denote by $\tilde{v}$ the new function. It is a nonnegative subsolution of (1.2) which tends to $\infty$ on $\partial \Omega$. For constructing a supersolution we recall that if $M\le -\mu ^*(1)$ there exist two types of explicit solutions of

$$\begin{aligned} -u''=u^p+M\left| u'\right| ^{\frac{2p}{p+1}} \end{aligned}$$

(2.26)

defined on $\mathbb {R}$ by $U_{j,M}(t)=\infty$ for $t\le 0$ and $U_{j,M}(t)=X_{j,M}t^{-\frac{2}{p-1}}$, j=1,2, for $t>0$ where $X_{1,M}$ and $X_{2,M}$ are respectively the smaller and the larger positive root of

$$\begin{aligned} X^{p-1}-\left| M\right| \left( {{\displaystyle \frac{2}{p-1} }}\right) ^{\frac{2}{p+1}} X^{\frac{p-1}{p+1}}+{\displaystyle \frac{2(p+1)}{(p-1)^2} }=0. \end{aligned}$$

(2.27)

Since $\Omega ^c$ is convex it is the intersection of all the closed half-spaces which contain it and we denote by ${\mathcal {H}}_\Omega$ the family of such hyperplanes which are touching $\partial \Omega$. If $H\in {\mathcal {H}}_\Omega$ let $\mathbf{n}_H$ be the normal direction to H, inward with respect to $\Omega$, ${\mathcal {H}}_+=\{x\in \mathbb {R}^N:\langle \mathbf{n}_H,x-\mathbf{n}_H\rangle >0\}$ and we define $U_H$ in the direction $\mathbf{n}_H$ by putting

$$\begin{aligned} U_H(x)=U_{2,M}(\langle \mathbf{n}_H,x-\mathbf{n}_H\rangle )=X_{2,M}\left( \langle \mathbf{n}_H,x-\mathbf{n}_H\rangle \right) ^{-\frac{2}{p-1}}\quad \text {for all }\,x\in {\mathcal {H}}_+. \end{aligned}$$

Hence and set, for $x\in \Omega :=\cap _{H\in {\mathcal {H}}_\Omega } {\mathcal {H}}_+$,

$$\begin{aligned} u_\Omega (x)=\inf _{H\in {\mathcal {H}}_\Omega }U_H(x). \end{aligned}$$

(2.28)

Then $u_\Omega$ is a nonnegative supersolution of (1.2) in $\Omega$ and

$$\begin{aligned} u_\Omega (x)\ge X_{2,M}(\mathrm{dist}\,x,\Omega ))^{-\frac{2}{p-1}}\qquad \forall x\in \Omega . \end{aligned}$$

Next $v_\Omega =\ln u_\Omega$ blows up on $\partial \Omega$, is finite on $\partial B_R$ and satisfies

$$\begin{aligned} -\Delta v_\Omega -ae^{(p-1)v_\Omega }\ge 0\qquad \text {in }B_R\cap \Omega . \end{aligned}$$

(2.29)

By comparison with w since $a<0$, $v_\Omega \ge w$. Hence $u_\Omega \ge v$ in $B_R{\setminus }\Omega ^c$. Extending v by zero as $\tilde{v}$ we obtain $u_\Omega \ge \tilde{v}$ in $\Omega ^c$. Hence $u_\Omega$ is a supersolution in $\Omega ^c$ where it dominates the subsolution $\tilde{v}$. It follows by [29, Theorem 1-4-6] that there exists a solution u of (1.2) satisfying $\tilde{v}\le u\le u_\Omega$, which ends the proof. $\square$

3 The refined Bernstein method

The method is a combination of the one used in the previous proofs. It is based upon the replacement of the unknown by setting first $u=v^{-\beta }$ as in [7] and [19] and the study of the equation satisfied by $\left| \nabla v\right|$. However we do not use integral techniques. Since u is a positive solution of (1.2) in $B_R$, the function v is well defined and satisfies

$$\begin{aligned} -\Delta v+(1+\beta )\frac{\left| \nabla v\right| ^2}{v}+\frac{1}{\beta }v^{1-\beta (p-1)}+M\left| \beta \right| ^{q-2}\beta v^{(\beta +1)(1-q)}\left| \nabla v\right| ^q=0 \end{aligned}$$

(3.1)

in $B_R$. We set

$$\begin{aligned} z=\left| \nabla v\right| ^2\,,\;s=1-q-\beta (q-1)=(1-q)(\beta +1)\,,\;\sigma =1-\beta (p-1), \end{aligned}$$

and derive

$$\begin{aligned} \Delta v=(1+\beta )\frac{z}{v}+\frac{1}{\beta }v^{\sigma }+M\left| \beta \right| ^{q-2}\beta v^{s}z^{\frac{q}{2}}. \end{aligned}$$

(3.2)

Combining Bochner’s formula and Schwarz identity we have classically

$$\begin{aligned} \frac{1}{2}\Delta z\ge \frac{1}{N}(\Delta v)^2+\langle \nabla \Delta v,\nabla v\rangle . \end{aligned}$$

We explicit the different terms

$$\begin{aligned} (\Delta v)^2= & {} (1+\beta )^2{\displaystyle \frac{z^2}{v^2} }+M^2\beta ^{2(q-1)}v^{2s}z^q+{\displaystyle \frac{v^{2\sigma }}{\beta ^2} } +2M(1+\beta )\left| \beta \right| ^{q-2}\beta v^{s-1}z^{1+\frac{q}{2}}\\&+{\displaystyle \frac{2(1+\beta )}{\beta } }v^{\sigma -1}z+2M\left| \beta \right| ^{q-2}v^{s+\sigma }z^{\frac{q}{2}},\\ \nabla \Delta v= & {} (1+\beta ){\displaystyle \frac{\nabla z}{v} }-{\displaystyle \frac{(1+\beta )z}{v^2} }\nabla v+{\displaystyle \frac{\sigma }{\beta } }v^{\sigma -1}\nabla v+Ms\left| \beta \right| ^{q-2}\beta v^{s-1}z^{\frac{q}{2}}\nabla v\\&+{\displaystyle \frac{Mq}{2} }\left| \beta \right| ^{q-2}\beta v^{s}z^{\frac{q}{2}-1}\nabla z,\\ \langle \nabla \Delta v,\nabla v\rangle= & {} \left( {\displaystyle \frac{1+\beta }{v} }+{\displaystyle \frac{Mq}{2} }\left| \beta \right| ^{q-2}\beta v^{s}z^{\frac{q}{2}-1}\right) \langle \nabla z,\nabla v\rangle -{\displaystyle \frac{(1+\beta )z^2}{v^2} }+{\displaystyle \frac{\sigma }{\beta } }v^{\sigma -1}z\\&+Ms\left| \beta \right| ^{q-2}\beta v^{s-1}z^{\frac{q}{2}+1}. \end{aligned}$$

Hence

$$\begin{aligned}&-{\displaystyle \frac{1}{2} }\Delta z+{\displaystyle \frac{1}{N} }(\Delta v)^2+\left( {\displaystyle \frac{1+\beta }{v} }+{\displaystyle \frac{Mq}{2} }\left| \beta \right| ^{q-2}\beta v^{s}z^{\frac{q}{2}-1}\right) \langle \nabla z,\nabla v\rangle \nonumber \\&\quad -{\displaystyle \frac{(1+\beta )z^2}{v^2} }+{\displaystyle \frac{\sigma }{\beta } }v^{\sigma -1}z+Ms\left| \beta \right| ^{q-2}\beta v^{s-1}z^{\frac{q}{2}+1}\le 0. \end{aligned}$$

(3.3)

3.1 Proof of Theorem D

We develop the term $(\Delta v)^2$ in (3.3) and get

$$\begin{aligned}&-{\displaystyle \frac{1}{2} }\Delta z +\left( {\displaystyle \frac{(1+\beta )^2}{N} }-(1+\beta )\right) {\displaystyle \frac{z^2}{v^2} }+{\displaystyle \frac{M^2\beta ^{2(q-1)}}{N} }v^{2s}z^q\nonumber \\&\qquad +M\left( s+{\displaystyle \frac{2(1+\beta )}{N} }\right) \left| \beta \right| ^{q-2}\beta v^{s-1}z^{1+\frac{q}{2}}\nonumber \\&\qquad +{\displaystyle \frac{v^{2\sigma }}{N\beta ^2} }+\left( {\displaystyle \frac{1+\beta }{v} }+{\displaystyle \frac{Mq}{2} }\left| \beta \right| ^{q-2}\beta v^{s}z^{\frac{q}{2}-1}\right) \langle \nabla z,\nabla v\rangle \nonumber \\&\qquad +{\displaystyle \frac{N\sigma +2(1+\beta )}{N\beta } }v^{\sigma -1}z+{\displaystyle \frac{2M\left| \beta \right| ^{q-2}}{N} }v^{s+\sigma }z^{\frac{q}{2}}\nonumber \\&\quad \le 0. \end{aligned}$$

(3.4)

Next we set $z=v^{-k}Y$ where k is a real parameter. Then $\nabla z=-kv^{-k-1}Y\nabla v+v^{-k}\nabla Y$,

$$\begin{aligned} \langle \nabla z,\nabla v\rangle= & {} -kv^{-k-1}Yz+v^{-k}\langle \nabla Y,\nabla v\rangle =-kv^{-2k-1}Y^2+v^{-k}\langle \nabla Y,\nabla v\rangle ,\\ {\displaystyle \frac{\langle \nabla z,\nabla v\rangle }{v} }= & {} -kv^{-2k-2}Y^2+v^{-k-1}\langle \nabla Y,\nabla v\rangle ,\\ Mv^sz^{\frac{q}{2}-1}\langle \nabla z,\nabla v\rangle= & {} -kMv^{s-\frac{qk}{2}-k-1}Y^{\frac{q}{2}+1}+Mv^{s-\frac{qk}{2}}Y^{\frac{q}{2}-1}\langle \nabla Y,\nabla v\rangle ,\\ -\Delta z= & {} \mathrm{div}\left( kv^{-k-1}Y\nabla v-v^{-k}\nabla Y\right) \\= & {} kv^{-k-1}Y\Delta v-k(k+1)v^{-k-2}Yz+2kv^{-k-1}\langle \nabla Y,\nabla v\rangle -v^{-k}\Delta Y\\= & {} kv^{-k-1}Y\Delta v-k(k+1)v^{-2k-2}Y^2+2kv^{-k-1}\langle \nabla Y,\nabla v\rangle -v^{-k}\Delta Y. \end{aligned}$$

From (3.2)

$$\begin{aligned} \Delta v=(1+\beta )v^{-k-1}Y+\frac{1}{\beta }v^{\sigma }+M\left| \beta \right| ^{q-2}\beta v^{s-k\frac{q}{2}}Y^{\frac{q}{2}}, \end{aligned}$$

therefore

$$\begin{aligned} -\Delta z= & {} k(\beta -k)v^{-2k-2}Y^2+{\displaystyle \frac{k}{\beta } }v^{\sigma -k-1}Y+kM\left| \beta \right| ^{q-2}\beta v^{s-k\frac{q}{2}-k-1}Y^{\frac{q}{2}+1}\\&+\,2kv^{-k-1}\langle \nabla Y,\nabla v\rangle -v^{-k}\Delta Y. \end{aligned}$$

Replacing $\langle \nabla z,\nabla v\rangle$ and $\Delta z$ given by the above expressions in (3.4) and z by $v^{-k}Y$, leads to

$$\begin{aligned}&-\Delta Y+\left( {\displaystyle \frac{k(\beta -k)}{2} }+{\displaystyle \frac{(1+\beta )^2}{N} }-(k+1)(\beta +1)\right) v^{-k-2}Y^2+{\displaystyle \frac{v^{2\sigma +k}}{N\beta ^2} }\\&\quad \quad \quad +{\displaystyle \frac{M^2\beta ^{2(q-1)}}{N} }v^{2s+k-kq}Y^q\\&\quad \quad \quad +\left( {\displaystyle \frac{k+\beta +1}{v} }+{\displaystyle \frac{Mq\left| \beta \right| ^{q-2}\beta }{2} } v^{s+k-k\frac{q}{2}}Y^{\frac{q}{2}-1}\right) \langle \nabla Y,\nabla v\rangle \\&\quad \quad \quad + {\displaystyle \frac{2M\left| \beta \right| ^{q-2}}{N} }v^{s+\sigma +k-k\frac{q}{2}}Y^{\frac{q}{2}}\\&\quad \quad \quad +\left( s+{\displaystyle \frac{2(1+\beta )}{N} }-{\displaystyle \frac{k(q-1)}{2} }\right) M\left| \beta \right| ^{q-2}\beta v^{s-k\frac{q}{2}-1}Y^{1+\frac{q}{2}}\\&\quad \quad \quad +{\displaystyle \frac{1}{\beta } }\left( {\displaystyle \frac{k}{2} }+\sigma +{\displaystyle \frac{2(1+\beta )}{N} }\right) v^{\sigma -1}Y\le 0. \end{aligned}$$

For $\epsilon _1,\epsilon _2>0$,

$$\begin{aligned}&{\displaystyle \frac{1}{v} }\left| \langle \nabla Y,\nabla v\rangle \right| \le \epsilon _1 v^{-k-2}Y^2+{\displaystyle \frac{1}{4\epsilon _1} }{\displaystyle \frac{\left| \nabla Y\right| ^2}{Y} },\\&\quad \quad v^{s+k-k\frac{q}{2}}Y^{\frac{q}{2}-1}\left| \langle \nabla Y,\nabla v\rangle \right| \le \epsilon _2v^{2s-kq+k}Y^{q}+{\displaystyle \frac{1}{4\epsilon _2} }{\displaystyle \frac{\left| \nabla Y\right| ^2}{Y} }. \end{aligned}$$

Hence

$$\begin{aligned}&-\Delta Y+{\displaystyle \frac{v^{2\sigma +k}}{N\beta ^2} }+ {\displaystyle \frac{2M\left| \beta \right| ^{q-2}}{N} }v^{s+\sigma +k -k\frac{q}{2}}Y^{\frac{q}{2}}+\left( {\displaystyle \frac{M^2\beta ^{2(q-1)}}{N} }-{\displaystyle \frac{Mq\epsilon _2\left| \beta \right| ^{q-1}}{2} }\right) v^{2s+k-kq}Y^q\nonumber \\&\qquad +\left( {\displaystyle \frac{k(\beta -k)}{2} }+{\displaystyle \frac{(1+\beta )^2}{N} }-(k+1)(\beta +1)-\left| k+\beta +1\right| \epsilon _1\right) v^{-k-2}Y^2\nonumber \\&\qquad +{\displaystyle \frac{1}{\beta } }\left( {\displaystyle \frac{k}{2} }+\sigma +{\displaystyle \frac{2(1+\beta )}{N} }\right) v^{\sigma -1}Y +\left( s+{\displaystyle \frac{2(1+\beta )}{N} }-{\displaystyle \frac{k(q-1)}{2} }\right) \nonumber \\&\qquad \times M\left| \beta \right| ^{q-2}\beta v^{s-k\frac{q}{2}-1}Y^{1+\frac{q}{2}}\nonumber \\&\quad \le \left( {\displaystyle \frac{\left| k+\beta +1\right| }{\epsilon _1} }+{\displaystyle \frac{Mq\left| \beta \right| ^{q-1}}{2\epsilon _2} }\right) {\displaystyle \frac{\left| \nabla Y\right| ^2}{4Y} }. \end{aligned}$$

(3.5)

We first choose $\epsilon _2=\frac{M\left| \beta \right| ^{q-1}}{qN}$, then

$$\begin{aligned}&-\Delta Y+{\displaystyle \frac{v^{2\sigma +k}}{N\beta ^2} }+\left( {\displaystyle \frac{k(\beta -k)}{2} }+{\displaystyle \frac{(1+\beta )^2}{N} } -(k+1)(\beta +1) -\left| k+\beta +1\right| \epsilon _1\right) v^{-k-2}Y^2\nonumber \\&\qquad +{\displaystyle \frac{1}{\beta } }\left( {\displaystyle \frac{k}{2} }+\sigma +{\displaystyle \frac{2(1+\beta )}{N} }\right) v^{\sigma -1}Y+ {\displaystyle \frac{M^2\beta ^{2(q-1)}}{2N} }v^{2s+k-kq}Y^q+ {\displaystyle \frac{2M\left| \beta \right| ^{q-2}}{N} }v^{s+\sigma +k-k\frac{q}{2}}Y^{\frac{q}{2}}\nonumber \\&\qquad +\left( s+{\displaystyle \frac{2(1+\beta )}{N} }-{\displaystyle \frac{k(q-1)}{2} }\right) M\left| \beta \right| ^{q-2}\beta v^{s-k\frac{q}{2}-1}Y^{1+\frac{q}{2}}\nonumber \\&\quad \le \left( {\displaystyle \frac{\left| k+\beta +1\right| }{\epsilon _1} }+{\displaystyle \frac{Nq^2}{2} }\right) {\displaystyle \frac{\left| \nabla Y\right| ^2}{4Y} }. \end{aligned}$$

(3.6)

In order to show the sign of the terms on the left in (3.5), we separate the terms containing the coefficient M from the ones which do not contain it. Indeed these last terms are associated to the mere Lane–Emden equation (1.3) which is treated, as a particular case, in [9, Theorem B] where the exponents therein are $q=0$, and $p\in \left( 1,\frac{N+3}{N-1}\right)$. We set

$$\begin{aligned} H_{\epsilon _1,1}&=\,{\displaystyle \frac{v^{2\sigma +k}}{N\beta ^2} }+\left( {\displaystyle \frac{k(\beta -k)}{2} } +{\displaystyle \frac{(1+\beta )^2}{N} }-(k+1)(\beta +1)-\left| k+\beta +1\right| \epsilon _1\right) v^{-k-2}Y^2\nonumber \\&\quad +{\displaystyle \frac{1}{\beta } }\left( {\displaystyle \frac{k}{2} }+\sigma +{\displaystyle \frac{2(1+\beta )}{N} }\right) v^{\sigma -1}Y\nonumber \\&=\,v^{2\sigma +k}\tilde{H}_{\epsilon _1,1}(v^{-1-k-\sigma }Y), \end{aligned}$$

(3.7)

where

$$\begin{aligned} \tilde{H}_{\epsilon _1,1}(t)&=\left( {\displaystyle \frac{k(\beta -k)}{2} }+{\displaystyle \frac{(1+\beta )^2}{N} }-(k+1) (\beta +1)-\left| k+\beta +1\right| \epsilon _1\right) t^2\nonumber \\&\quad +{\displaystyle \frac{1}{\beta } }\left( {\displaystyle \frac{k}{2} }+\sigma +{\displaystyle \frac{2(1+\beta )}{N} }\right) t +{\displaystyle \frac{1}{N\beta ^2} }, \end{aligned}$$

(3.8)

and

$$\begin{aligned} H_{M,2}&={\displaystyle \frac{M^2\beta ^{2(q-1)}}{2N} }v^{2s+k-kq}Y^q+{\displaystyle \frac{2M\left| \beta \right| ^{q-2}}{N} } v^{s+\sigma +k-k\frac{q}{2}}Y^{\frac{q}{2}}\nonumber \\&\quad +\,\left( s+{\displaystyle \frac{2(1+\beta )}{N} }-{\displaystyle \frac{k(q-1)}{2} }\right) M\left| \beta \right| ^{q-2}\beta v^{s-k\frac{q}{2}-1}Y^{1+\frac{q}{2}}. \end{aligned}$$

(3.9)

Then

$$\begin{aligned} -\Delta Y+v^{2\sigma +k}\tilde{H}_{\epsilon _1,1}(v^{-1-k-\sigma }Y)+H_{M,2}\le \left( {\displaystyle \frac{\left| k+\beta +1\right| }{\epsilon _1} }+{\displaystyle \frac{Nq^2}{2} }\right) {\displaystyle \frac{\left| \nabla Y\right| ^2}{4Y} }. \end{aligned}$$

The sign of $\tilde{H}_{\epsilon _1,1}$ depends on its discriminant ${\mathcal {D}}_{\epsilon _1}$ which is a polynomial in its coefficients. Then if for $\epsilon _1=0$ this discriminant is negative ${\mathcal {D}}_{0}$ is negative, the discriminant ${\mathcal {D}}_{\epsilon _1}$ of $\tilde{H}_{\epsilon _1,1}$ shares this property for $\epsilon _1>0$ small enough and therefore $H_{\epsilon _1,1}$ is positive. The proof is similar as the one of [9, Theorem B] in case (i) but for the sake of completeness we recall the main steps. Firstly

$$\begin{aligned} {\mathcal {D}}'_0:= & {} N^2\beta ^2{\mathcal {D}}_0=\left( {\displaystyle \frac{Nk}{2} }+\sigma N+2(1+\beta )\right) ^2\\&-\,4\left( {\displaystyle \frac{Nk(\beta -k)}{2} }+(1+\beta )^2-N(k+1)(\beta +1)\right) . \end{aligned}$$

Then

$$\begin{aligned} {\mathcal {D}}'_0=\left( {\displaystyle \frac{N(p-1)}{4} }-1\right) (2\sigma +k)^2+2(p-1)(2\sigma +k)+\tilde{L} \end{aligned}$$

where $\tilde{L}=(p-1)k^2+p(\lambda +2)^2>0$. Put

$$\begin{aligned} S= & {} {\displaystyle \frac{2\sigma +k}{k+2} }=1-{\displaystyle \frac{2\beta (p-1)}{k+2} }\,\text { and } \\ {\mathcal {T}}(S)= & {} \left( {\displaystyle \frac{(N-1)(p-1)}{4} }-1\right) S^2+(p-1)S+p. \end{aligned}$$

After some computations we get, if $k\ne -2$,

$$\begin{aligned} {\mathcal {D}}'_1:={\displaystyle \frac{(p-1){\mathcal {D}}'_0}{(k+2)^2} }=(p-1)\left( {\displaystyle \frac{k}{k+2} } -{\displaystyle \frac{S}{2} }\right) ^2 +{\mathcal {T}}(S). \end{aligned}$$

(3.10)

We choose $S>2$ such that $\frac{k}{k+2}-\frac{S}{2}=0$, hence $\beta =\frac{2-k}{2(p-1)}$. If $p<\frac{N+3}{N-1}$ the coefficient of $S^2$ in ${\mathcal {T}}(S)$ is negative. Hence ${\mathcal {T}}(S)<0$ provided S is large enough which is satisfied if $k<-2$ with $\left| k+2\right|$ small enough. We infer from this that $\beta >0$, ${\mathcal {D}}_0<0$ and $\tilde{H}_{\epsilon _1,1}>0$ if $\epsilon _1$ is small enough. In particular $\tilde{H}_{\epsilon _1,1}(t)\ge c_6 (t^2+1)$ for some $c_6=c_6(N,p,q)>0$, which means

$$\begin{aligned} v^{2\sigma +k}\tilde{H}_{\epsilon _1,1}(v^{-1-k-\sigma }Y)\ge c_6\left( v^{-k-2}Y^2+v^{2\sigma +k}\right) . \end{aligned}$$

(3.11)

Secondly the positivity of $H_{M,2}$ is ensured, as $\beta$ and M are positive, by the positivity of

$$\begin{aligned} {\mathcal {A}}:=s+{\displaystyle \frac{2(1+\beta )}{N} }-{\displaystyle \frac{k(q-1)}{2} }. \end{aligned}$$

Replacing s by its value, we obtain, since $1<q<\frac{N+2}{N}$ and $\beta +\frac{2+k}{2}>0$, which can be assume by taking $\left| k+2\right|$ small enough,

$$\begin{aligned} {\mathcal {A}}=2{\displaystyle \frac{1+\beta }{N} }-(q-1)\left( \beta +1+{\displaystyle \frac{k}{2} }\right) >-{\displaystyle \frac{k}{N} } \end{aligned}$$

Then we deduce that

$$\begin{aligned} -\Delta Y +c_6\left( v^{-k-2}Y^2+v^{2\sigma +k}\right) \le c_7{\displaystyle \frac{\left| \nabla Y\right| ^2}{Y} }, \end{aligned}$$

(3.12)

and $c_7=c_7(N,p,q)>0$ is independent of M. Since $S=1-\frac{2\beta (p-1)}{k+2}=1-\frac{2-k}{k+2}=\frac{2k}{k+2}>0$, we have

$$\begin{aligned} 2Y^{\frac{2S}{S+1}}=2\left( {\displaystyle \frac{Y^2}{v^{k+2}} }\right) ^{\frac{S}{S+1}}v^{\frac{(k+2)S}{S+1}}\le {\displaystyle \frac{Y^2}{v^{k+2}} }+v^{(k+2)S}={\displaystyle \frac{Y^2}{v^{k+2}} }+v^{2\sigma +k}. \end{aligned}$$

(3.13)

From this we infer the inequality

$$\begin{aligned} -\Delta Y +2c_6Y^{\frac{2S}{S+1}}\le c_7{\displaystyle \frac{\left| \nabla Y\right| ^2}{Y} }. \end{aligned}$$

(3.14)

Then we derive from Lemma 2.2 that in the ball $B_R$ there holds

$$\begin{aligned} Y(0)\le c_8R^{-\frac{2(S+1)}{S-1}}=c_8R^{-2+\frac{2(k+2)}{\beta (p-1)}}. \end{aligned}$$

(3.15)

From this it follows

$$\begin{aligned} \left| \nabla u^{-\frac{2+k}{2\beta }}(0)\right| \le {\displaystyle \frac{\left| k+2\right| }{2} }\sqrt{c_8\,}R^{-1+\frac{k+2}{\beta (p-1)}}. \end{aligned}$$

(3.16)

Setting $a=-\frac{k+2}{2\beta }>0$ we get that for any domain $\Omega \subset \mathbb {R}^N$ any positive solution in $\Omega$ satisfies

$$\begin{aligned} \left| \nabla u^{a}(x)\right| \le {\displaystyle \frac{|k+2|}{2} }\sqrt{c_8\,}\left( \mathrm{dist}\,(x,\partial \Omega ) \right) ^{-1-\frac{2a}{p-1}}\qquad \text {for all }\, x\in \Omega . \end{aligned}$$

(3.17)

The non existence of any positive of (1.2) solution in $\mathbb {R}^N$ follows classically. $\square$

Corollary 3.1

Let $\Omega$ be a smooth domain in $\mathbb {R}^N$, $N \ge 2$ with a bounded boundary, $1<p<\frac{N+3}{N-1}$, $1<q<\frac{N+2}{N}$ and $M>0$. If u is a positive solution of (1.2) in $\Omega$ there exists $d_0$ depending on $\Omega$ and $c_9 = c_9(N,p,q) > 0$ such that

$$\begin{aligned} u(x)\le c_9\left( \left( \mathrm{dist}\,(x,\partial \Omega )\right) ^{-\frac{2}{p-1}} +\max _{\mathrm{dist}\,(z,\partial \Omega )=d_0}u(z)\right) \qquad \text {for all }\, x\in \Omega . \end{aligned}$$

(3.18)

Proof

It is similar to the one of [9, Corollary B-2]. $\square$

4 The integral method

4.1 Preliminary inequalities

We recall the next inequality [6, Lemma 3.1].

Lemma 4.1

Let $\Omega \subset \mathbb {R}^N$ be a domain. Then for any positive $u\in C^2(\Omega )$, any nonnegative $\eta \in C^\infty _0(\Omega )$ and any real numbers m and d such that $d\ne m+2$, the following inequality holds

$$\begin{aligned} A{\displaystyle \int _{\Omega }^{}}\eta u^{m-2}\left| \nabla u\right| ^4 dx-{\displaystyle \frac{N-1}{N} }{\displaystyle \int _{\Omega }^{}}\eta u^{m}(\Delta u)^2 dx -B{\displaystyle \int _{\Omega }^{}}\eta u^{m-1}\left| \nabla u\right| ^2 \Delta u dx\le R, \end{aligned}$$

(4.1)

where

$$\begin{aligned} A={\displaystyle \frac{1}{4N} }\left( 2(N-m)d-(N-1)(m^2+d^2)\right) \,,\; B={\displaystyle \frac{1}{2N} }\left( 2(N-1)m+(N+2)d\right) , \end{aligned}$$

and

$$\begin{aligned} R={\displaystyle \frac{m+d}{2} }{\displaystyle \int _{\Omega }^{}} u^{m-1}\left| \nabla u\right| ^2\langle \nabla u,\nabla \eta \rangle dx +{\displaystyle \int _{\Omega }^{}}u^{m}\Delta u \langle \nabla u,\nabla \eta \rangle dx+ {\displaystyle \frac{1}{2} }{\displaystyle \int _{\Omega }^{}}u^{m}\left| \nabla u\right| ^2\Delta \eta dx. \end{aligned}$$

It is noticeable that d is a free parameter which plays a role only in the coefficients of the integral terms. The following technical result is useful to deal with the multi-parameter constraints problems which occur in our construction. It was first used in [7] under a simpler form and extended in [6, Lemma 3.4].

Lemma 4.2

For any $N\in \mathbb {N}$, $N\ge 3$ and $1<p<\frac{N+2}{N-2}$ there exist real numbers m and d verifying

$$\begin{aligned} \begin{array}{ll} \mathrm{(i)} &{}\quad d\ne m+2,\\ \mathrm{(ii)} &{}\quad {\displaystyle \frac{2(N-1)p}{N+2} }<d,\\ \mathrm{(iii)} &{}\quad \max \left\{ -2,1-p,{\displaystyle \frac{(N-4)p-N}{2} }\right\} <m\le 0,\\ \mathrm{(iv)}&{} \quad 2(N-m)d-(N-1)(m^2+d^2)>0. \end{array} \end{aligned}$$

(4.2)

4.2 Proof of Theorem E

Step 1: The integral estimates. Let $\eta \in C^{\infty }_0(\Omega )$, $\eta \ge 0$. We apply Lemma 4.1 to a positive solution $u\in C^2(\Omega )$ of (1.2), firstly with $q>1$ and then with $q=\frac{2p}{p+1}$.

$$\begin{aligned}&A{\displaystyle \int _{\Omega }^{}}\eta u^{m-2}\left| \nabla u\right| ^4 dx-{\displaystyle \frac{N-1}{N} }{\displaystyle \int _{\Omega }^{}}\eta \left( u^{m+2p}+2Mu^{m+p}\left| \nabla u\right| ^q+M^2u^{m}\left| \nabla u\right| ^{2q}\right) dx\nonumber \\&\quad -B{\displaystyle \int _{\Omega }^{}}\eta u^{m-1}\left| \nabla u\right| ^2 \Delta u dx\le R.\quad \end{aligned}$$

(4.3)

We multiply (1.2) by $\eta u^{m+p}$ and integrate over $\Omega$. Then

$$\begin{aligned}&{\displaystyle \int _{\Omega }^{}}\eta \left( u^{m+2p}+Mu^{m+p}\left| \nabla u\right| ^q \right) dx=-{\displaystyle \int _{\Omega }^{}}\eta u^{m+p}\Delta udx\\&\quad ={\displaystyle \int _{\Omega }^{}}u^{m+p}\langle \nabla u,\nabla \eta \rangle dx+(m+p){\displaystyle \int _{\Omega }^{}}\eta u^{m+p-1}\left| \nabla u\right| ^2 dx. \end{aligned}$$

We set

$$\begin{aligned} F= & {} {\displaystyle \int _{\Omega }^{}}\eta u^{m-2}\left| \nabla u\right| ^4 dx\,,\;P={\displaystyle \int _{\Omega }^{}}\eta u^{m-1}\left| \nabla u\right| ^{q+2} dx\,,\;V={\displaystyle \int _{\Omega }^{}}\eta u^{m+2p} dx,\\ T= & {} {\displaystyle \int _{\Omega }^{}}\eta u^{m+p-1}\left| \nabla u\right| ^{2} dx\,,\;W={\displaystyle \int _{\Omega }^{}}\eta u^{m+p}\left| \nabla u\right| ^{q} dx\,,\;U={\displaystyle \int _{\Omega }^{}}\eta u^{m} \left| \nabla u\right| ^{2q}dx,\\ S= & {} {\displaystyle \int _{\Omega }^{}} u^{m+p}\langle \nabla u,\nabla \eta \rangle dx, \end{aligned}$$

so that there holds

$$\begin{aligned} AF-{\displaystyle \frac{N-1}{N} }\left( V+2MW+M^2U\right) +BT+BMP\le R, \end{aligned}$$

(4.4)

and

$$\begin{aligned} V+MW=(m+p)T+S. \end{aligned}$$

(4.5)

Eliminating V between (4.4) and (4.5), we get

$$\begin{aligned} AF+B_0T+M\left( BP-{\displaystyle \frac{N-1}{N} }W-{\displaystyle \frac{N-1}{N} }MU\right) \le R-{\displaystyle \frac{N-1}{N} }S, \end{aligned}$$

(4.6)

where

$$\begin{aligned} B_0=B-{\displaystyle \frac{N-1}{N} }(m+p)={\displaystyle \frac{N+2}{2N} }d-{\displaystyle \frac{N-1}{N} }p. \end{aligned}$$

Also

$$\begin{aligned} 2P=2{\displaystyle \int _{\Omega }^{}}\eta u^{m}{\displaystyle \frac{\left| \nabla u\right| ^{2}}{u} }\left| \nabla u\right| ^{q} dx\le {\displaystyle \int _{\Omega }^{}}\eta u^m\left( {\displaystyle \frac{\left| \nabla u\right| ^{4}}{u^2} }+\left| \nabla u\right| ^{2q}\right) dx=F+U. \end{aligned}$$

We fix now $q=\frac{2p}{p+1}$, then

$$\begin{aligned} U&={\displaystyle \int _{\Omega }^{}}\eta u^m\left| \nabla u\right| ^{2q}dx={\displaystyle \int _{\Omega }^{}}\eta u^m\left( {\displaystyle \frac{\left| \nabla u\right| }{\sqrt{u}} }\right) ^{4(q-1)}u^{2(q-1)}\left| \nabla u\right| ^{4-2q}dx\nonumber \\&\le {\displaystyle \frac{p-1}{p+1} }{\displaystyle \int _{\Omega }^{}}\eta u^{m-2}\left| \nabla u\right| ^{4}dx+{\displaystyle \frac{2}{p+1} }{\displaystyle \int _{\Omega }^{}}\eta u^{m+p-1}\left| \nabla u\right| ^{2}dx\nonumber \\&\le {\displaystyle \frac{p-1}{p+1} }F+{\displaystyle \frac{2}{p+1} }T, \end{aligned}$$

(4.7)

hence

$$\begin{aligned} P\le {\displaystyle \frac{1}{2} }F+{\displaystyle \frac{1}{2} }U\le {\displaystyle \frac{p}{p+1} }F+{\displaystyle \frac{1}{p+1} }T \end{aligned}$$

(4.8)

and

$$\begin{aligned} 2W=&\,2{\displaystyle \int _{\Omega }^{}}\eta u^{m+p}\left| \nabla u\right| ^{q}dx\le {\displaystyle \int _{\Omega }^{}}\eta u^{m+2p}dx+{\displaystyle \int _{\Omega }^{}}\eta u^{m}\left| \nabla u\right| ^{2q}dx=V+U\nonumber \\ \le&\, U+(m+p)T+S-MW. \end{aligned}$$

(4.9)

Next we assume that $|M|\le 1$. From (4.7), (4.9), it follows that

$$\begin{aligned} W\le U+(m+p)T+S\le F+(m+p+1)T+S. \end{aligned}$$

(4.10)

From now we fix m and d according Lemma 4.2. Therefore $A>0$ by (4.2)-(iv) and $B>0$ by combining (4.2)-(ii) and (4.2)-(iii). Furthermore $B_0>0$ by (4.2)-(ii). Hence, from (4.7), (4.8) and (4.10) we derive, since $\frac{N-1}{N}<1$ and $m\le 0$ from (4.2)-(ii)

$$\begin{aligned} \left| BP-{\displaystyle \frac{N-1}{N} }W-{\displaystyle \frac{N-1}{N} }MU\right|\le & {} B\left( F+T\right) +F+(p+1)T+S+F+T,\\\le & {} \left( B+2\right) F+\left( B+p+2\right) T+S. \end{aligned}$$

Plugging these estimates into (4.6) we infer

$$\begin{aligned} AF+B_0T-|M|\left( \left( B+2\right) F+\left( B+p+2\right) T+S\right) \le R-{\displaystyle \frac{N-1}{N} }S. \end{aligned}$$

(4.11)

Since A and $B_0$ are positive, there exists $\mu _1\in (0,1)$ such that for any $\left| M\right| <\mu _1$,

$$\begin{aligned} A_1:=A-|M|\left( B+2\right)>\frac{A}{2}\quad \text {and }\; B_1:=B_0-\left| M\right| \left( B+p+2\right) >\frac{B_0}{2}. \end{aligned}$$

Set $A_2=\min \{A_1,B_1\}$, then, and whatever is the sign of S,

$$\begin{aligned} A_2(F+T)\le \left| R\right| +\left| S\right| . \end{aligned}$$

Using (4.7) and (4.8) we have

$$\begin{aligned} A_2(U+P)\le 2A_2(F+T)\le 2(|R|+|S|). \end{aligned}$$

(4.12)

In the sequel we denote by $c_j$ some positive constants depending on N and p. Then

$$\begin{aligned} U+P+F+T+W\le c_{1}(|R|+|S|). \end{aligned}$$

(4.13)

On the other hand, we have

$$\begin{aligned} \left| R\right| \le c_{2}{\displaystyle \int _{\Omega }^{}} \left( u^{m-1}\left| \nabla u\right| ^3\left| \nabla \eta \right| +u^{m+p}\left| \nabla u\right| \left| \nabla \eta \right| +u^{m}\left| \nabla u\right| ^{q+1}\left| \nabla \eta \right| +u^{m}\left| \nabla u\right| ^2\left| \Delta \eta \right| \right) dx. \end{aligned}$$

Since

$$\begin{aligned} \left| \nabla u\right| ^q=\left( {\displaystyle \frac{\left| \nabla u\right| }{\sqrt{u}} }\right) ^qu^{\frac{q}{2}}\le {\displaystyle \frac{\left| \nabla u\right| ^2}{ u} }+ u^{\frac{q}{2-q}}={\displaystyle \frac{\left| \nabla u\right| ^2}{ u} }+ u^{p}, \end{aligned}$$

we deduce

$$\begin{aligned} {\displaystyle \int _{\Omega }^{}} u^m|\nabla u|^{q+1}|\nabla \eta |dx\le {\displaystyle \int _{\Omega }^{}} u^{m-1}|\nabla u|^{3}|\nabla \eta |dx +{\displaystyle \int _{\Omega }^{}} u^{m+p}|\nabla u||\nabla \eta |dx. \end{aligned}$$

Thus we derive from (4.13)

$$\begin{aligned} U+P+F+T+W\le & {} 2c_{3}\left( {\displaystyle \int _{\Omega }^{}} u^{m-1}|\nabla u|^{3}|\nabla \eta |dx+{\displaystyle \int _{\Omega }^{}} u^{m+p}|\nabla u||\nabla \eta |dx\right. \nonumber \\&\left. +{\displaystyle \int _{\Omega }^{}} u^{m}\left| \nabla u\right| ^2\left| \Delta \eta \right| dx \right) . \end{aligned}$$

(4.14)

From this point we can use the method developed in [7, p 599] for proving the Harnack inequality satisfied by positive solutions of (1.3) in $\Omega$. We set $\eta =\xi ^\lambda$ with $\xi \in C^\infty _0(\Omega )$ with value in [0, 1] and $\lambda >4$. For $\epsilon \in (0,1)$ we have by the Hölder–Young inequality

$$\begin{aligned}&{\displaystyle \int _{\Omega }^{}} u^{m-1}|\nabla u|^{3}|\nabla \xi ^\lambda |dx\le {\displaystyle \frac{\epsilon }{4c_{3}} }{\displaystyle \int _{\Omega }^{}} u^{m-2}|\nabla u|^{4}\xi ^\lambda dx +C(\epsilon ,c_3){\displaystyle \int _{\Omega }^{}} u^{m+2}|\nabla \xi |^{4}\xi ^{\lambda -4}dx,\nonumber \\ \end{aligned}$$

(4.15)

$$\begin{aligned}&{\displaystyle \int _{\Omega }^{}} u^{m+p}|\nabla u||\nabla \xi ^p|dx\le {\displaystyle \frac{\epsilon }{4c_{3}} }{\displaystyle \int _{\Omega }^{}} u^{m+p-1}|\nabla u|^{2}\xi ^pdx +C(\epsilon ,c_3){\displaystyle \int _{\Omega }^{}} u^{m+p+1}|\nabla \xi |^{2}\xi ^{\lambda -2}dx,\nonumber \\ \end{aligned}$$

(4.16)

and

$$\begin{aligned} {\displaystyle \int _{\Omega }^{}} u^{m}|\nabla u|^2|\Delta \xi ^p|dx\le & {} {\displaystyle \frac{\epsilon }{4c_{3}} }{\displaystyle \int _{\Omega }^{}} u^{m-2}|\nabla u|^{4}\xi ^pdx \nonumber \\&+C(\epsilon ,c_3){\displaystyle \int _{\Omega }^{}} u^{m+2}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\right) \xi ^{\lambda -4} dx. \end{aligned}$$

(4.17)

Hence

$$\begin{aligned} U+P+F+T+W\le & {} c_{4} \left( {\displaystyle \int _{\Omega }^{}} u^{m+2}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\xi ^{2}\right) \xi ^{\lambda -4} dx\right. \nonumber \\&\left. +\,{\displaystyle \int _{\Omega }^{}} u^{m+p+1}|\nabla \xi |^{2}\xi ^{\lambda -2}dx \right) . \end{aligned}$$

(4.18)

Let us denote by $c_{4}X$ the right-hand side of (4.18). Combining (4.5), (4.16) and (4.18) we also get

$$\begin{aligned} S:= {\displaystyle \int _{\Omega }^{}} u^{m+p}|\nabla u||\nabla \xi ^p|dx\le c_{5}X\Longrightarrow V:={\displaystyle \int _{\Omega }^{}} u^{m+2p}\xi ^pdx\le c_{6}X, \end{aligned}$$

(4.19)

and we finally obtain

$$\begin{aligned} U+V+P+F+S+T+W\le c_{7} X. \end{aligned}$$

(4.20)

Finally we estimate the different terms in X, using that $m+p>0$ from (4.2)-(iii). For $\epsilon >0$

$$\begin{aligned}&{\displaystyle \int _{\Omega }^{}} u^{m+2}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\xi ^{2}\right) \xi ^{\lambda -4} dx\le \epsilon {\displaystyle \int _{\Omega }^{}} u^{m+2p}\xi ^{\lambda } dx\nonumber \\&\quad +C(\epsilon ,c_7){\displaystyle \int _{\Omega }^{}}\xi ^{\lambda -2\frac{m+2p}{p-1}}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\right) ^{\frac{m+2p}{2(p-1)}}dx, \end{aligned}$$

(4.21)

and

$$\begin{aligned} {\displaystyle \int _{\Omega }^{}} u^{m+p+1}|\nabla \xi |^{2}\xi ^{\lambda -2}dx\le \epsilon {\displaystyle \int _{\Omega }^{}} u^{m+2p}\xi ^{\lambda } dx +C(\epsilon ,c_7){\displaystyle \int _{\Omega }^{}}\xi ^{\lambda -2\frac{m+2p}{p-1}} |\nabla \xi |^{\frac{2(m+2p)}{p-1}}dx. \end{aligned}$$

(4.22)

At end we obtain

$$\begin{aligned} U+V+P+F+S+T+W\le c_{8} {\displaystyle \int _{\Omega }^{}}\xi ^{\lambda -2\frac{m+2p}{p-1}}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\right) ^{\frac{m+2p}{2(p-1)}}dx.\quad \end{aligned}$$

(4.23)

Step 2: The Harnack inequality. We suppose that $\Omega =B_R{\setminus }\{0\}:=B_R^*$, fix $y\in B_{\frac{R}{2}}^*$, set $r=|y|$, then $B_r(y)\subset B_R^*$. Let $\xi \in C^\infty _0(B_r(y))$ such that $0\le \xi \le 1$, $\xi =1$ in $B_{\frac{r}{2}}(y)$, $\left| \nabla \xi \right| \le cr^{-1}$ and $\left| \Delta \xi \right| \le cr^{-2}$. We choose $\lambda >\max \left\{ 4,\frac{m+2p}{p+1}\right\}$, then

$$\begin{aligned} {\displaystyle \int _{B_r(y)}^{}}\xi ^{\lambda -2\frac{m+2p}{p-1}}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\right) ^{\frac{m+2p}{2(p-1)}}dx \le c_{9}r^{N-\frac{2(m+2p)}{p-1}}, \end{aligned}$$

and hence

$$\begin{aligned} {\displaystyle \int _{B_{\frac{r}{2}}(y)}^{}}u^{m+2p}dx\le V\le c_{10}r^{N-\frac{2(m+2p)}{p-1}}. \end{aligned}$$

(4.24)

We write (1.2) under the form

$$\begin{aligned} \Delta u+D(x)u+M\langle G(x).\nabla u\rangle =0, \end{aligned}$$

(4.25)

with

$$\begin{aligned} D(x)=u^{p-1}\quad \text {and }\;\,G(x)=|\nabla u|^{-\frac{2}{p+1}}\nabla u. \end{aligned}$$

Set $\sigma =\frac{m+2p}{p-1}$, then $\sigma >\frac{N}{2}$ by (4.2)-(iii) and

$$\begin{aligned} {\displaystyle \int _{B_{\frac{r}{2}}(y)}^{}}D^{\sigma }(x)dx\le V\le c_{10}r^{N-\frac{2(m+2p)}{p-1}}=c_{10}r^{N-2\sigma }. \end{aligned}$$

(4.26)

Next we estimate G. For $\tau ,\omega ,\gamma >0$ and $\theta >1$, we have with $\theta '=\frac{\theta }{\theta -1}$,

$$\begin{aligned} \left| \nabla u\right| ^{(q-1)\tau }=u^{\omega }\left| \nabla u\right| ^{\gamma }u^{-\omega }\left| \nabla u\right| ^{(q-1)\tau -\gamma }\le u^{\omega \theta '}\left| \nabla u\right| ^{\gamma \theta }+u^{-\omega \theta }\left| \nabla u\right| ^{((q-1)\tau -\gamma )\theta '}. \end{aligned}$$

We fix

$$\begin{aligned} \tau =2{\displaystyle \frac{2p+m}{p-1} }=2\sigma \,,\; \omega ={\displaystyle \frac{(2-m)(p+m-1)}{p+1} }\quad \text {and}\quad \theta ={\displaystyle \frac{p+1}{2-m} }. \end{aligned}$$

Then $\omega >0$ and $\theta >1$ from (4.2)-(iii), $\omega >0$. Then $u^{\omega \theta '}\left| \nabla u\right| ^{\gamma \theta }=u^{p+m-1}\left| \nabla u\right| ^{2}$ and $u^{-\omega \theta }\left| \nabla u\right| ^{((q-1)\tau -\gamma )\theta '}=u^{m-2}\left| \nabla u\right| ^{4}$, thus

$$\begin{aligned} {\displaystyle \int _{B_{\frac{r}{2}}(y)}^{}}\left| \nabla u\right| ^{(q-1)\tau }dx\le F+T\le c_{11} {\displaystyle \int _{\Omega }^{}}\xi ^{\lambda -2\frac{m+2p}{p-1}}\left( |\nabla \xi |^{4}+\left| \Delta \xi \right| ^2\xi ^{2}\right) ^{\frac{m+2p}{2(p-1)}}dx. \end{aligned}$$

This implies

$$\begin{aligned} {\displaystyle \int _{B_{\frac{r}{2}}(y)}^{}}G^{\tau }(x)dx\le c_{12}r^{N-\tau }, \end{aligned}$$

(4.27)

with $\tau >N$. Using the results of [28, Section 5], we infer that a Harnack inequality, uniform with respect to r, is satisfied. Hence there exists $c_{13}>0$ depending on N, p such that for any $r\in (0,\frac{R}{2}]$ and y such that $\left| y\right| =r$ there holds

$$\begin{aligned} \max _{z\in B_{\frac{r}{2}}(y)}u(z)\le c_{13}\min _{z\in B_{\frac{r}{2}}(y)}u(z)\quad \forall 0<r\le \tfrac{R}{2}\;\;\forall y\,\text { s.t. }\,|y|=r, \end{aligned}$$

(4.28)

which implies

$$\begin{aligned} u(x)\le c_{14} u(x')\quad \forall x,x'\in \mathbb {R}^N\;\;\text { s.t. }\,|x|=|x'|\le \frac{R}{2}, \end{aligned}$$

(4.29)

and actually $c_{14}=c_{13}^{7}$ by a simple geometric construction. By (4.24)

$$\begin{aligned} r^N\omega _Nr^N\left( \min _{z\in B_{\frac{r}{2}}(y)}u(z)\right) ^{m+2p}\le 4^Nc_{10}r^{N-\frac{2(m+2p)}{p-1}}, \end{aligned}$$

where $\omega _N$ is the volume of the unit N-ball. This implies

$$\begin{aligned} u(x)\le c_{14}\left| x\right| ^{-\frac{2}{p-1}}\qquad \forall x\in B_{\frac{R}{2}}^*. \end{aligned}$$

(4.30)

The proof follows. $\square$

Remark

Using standard rescaling techniques (see e.g. [29, Lemma 3.3.2]) the gradient estimate holds

$$\begin{aligned} \left| \nabla u(x)\right| \le c_{15}\left| x\right| ^{-\frac{p+1}{p-1}}\qquad \forall x\in B_{\frac{R}{3}}^*. \end{aligned}$$

(4.31)

And the next estimate for a solution u in a domain $\Omega$ satisfying the interior sphere condition with radius R is valid

$$\begin{aligned} u(x)\le c_{14}\left( \mathrm{dist}\,( x,\partial \Omega )\right) ^{-\frac{2}{p-1}}\quad \forall x\in \Omega \;\text { s.t.}\; \mathrm{dist}\,( x,\partial \Omega )\le \frac{R}{2}. \end{aligned}$$

(4.32)

5 Radial ground states

We recall that if $q\ne \frac{2p}{p+1}$ and $M\ne 0$, (1.2) can be reduced to the case $M=\pm 1$ by using the transformation (1.15). Since any ground state u of (1.2) radial with respect to 0 is decreasing (this is classical and straightforward), it achieves its maximum at 0 and the following equivalence holds if v is defined by (1.15)

$$\begin{aligned} \begin{array}{lll} -u''-{\displaystyle \frac{N-1}{r} }u'=|u|^{p-1}u+M\left| u_r\right| ^q&{}\quad \text {s.t. }\, &{}\max u=u(0)=1\\ \Longleftrightarrow \\ -v''-{\displaystyle \frac{N-1}{r} }v'=|v|^{p-1}v\pm \left| v_r\right| ^q&{}\quad \text {s.t. }\, &{}\max v=v(0)=|M|^{\frac{2}{(p+1)q-2p}}. \end{array} \end{aligned}$$

(5.1)

Hence large or small values of M for u are exchanged into large or small values of v(0) for v and in the sequel we will essentially express our results using the function u.

5.1 Energy functions

We consider first the energy function

$$\begin{aligned} r\mapsto H(r)={\displaystyle \frac{u^{p+1}}{p+1} }+{\displaystyle \frac{u'^{2}}{2} }. \end{aligned}$$

(5.2)

Then

$$\begin{aligned} H'(r)=M\left| u'\right| ^{q+1}-{\displaystyle \frac{N-1}{r} }u'^{2}. \end{aligned}$$

Hence, if $M\le 0$, H is decreasing, a property often used in [25]. This implies in particular that a radial ground state satisfies

$$\begin{aligned} \left| u'(r)\right| \le \sqrt{{\displaystyle \frac{2}{p+1} }}\left( u(0)\right) ^{\frac{p+1}{2}}. \end{aligned}$$

(5.3)

A similar estimate holds in all the cases.

Proposition 5.1

Let $M>0$, $p,q>1$. If u is a radial ground state solution of (1.2), then the function H defined in (5.2) is decreasing and in particular (5.3) holds.

Proof

Let u be such a radial ground state. By Proposition 2.1 we must have $q>\frac{N}{N-1}$ and

$$\begin{aligned} {\displaystyle \frac{r}{u'^{2}} }H'=Mr\left| u'\right| ^{q-1}+1-N\le {\displaystyle \frac{(N-1)q-N}{q-1} }+1-N=-\frac{1}{q-1}, \end{aligned}$$

this implies the claim. $\square$

5.1.1 Exponential perturbations

As we have seen it in the introduction, if $q<\frac{2p}{p+1}$ Eq. (1.2) can be seen as a perturbation of the Lane–Emden equation (1.3) while if $q>\frac{2p}{p+1}$ it can be seen as a perturbation of the Ricatti equation (1.14). Two types of transformations can emphasize these aspects.

(1) For $p>1$ set

$$\begin{aligned} u(r)=r^{-\frac{2}{p-1}}x(t),\quad u'(r)=-r^{-\frac{p+1}{p-1}}y(t),\quad t=\ln r, \end{aligned}$$

(5.4)

then

$$\begin{aligned} \begin{aligned} x_t&=\frac{2}{p-1}x-y\\ y_t&=-Ky+x^p+Me^{-\omega t}y^q \end{aligned} \end{aligned}$$

(5.5)

with

$$\begin{aligned} K=\frac{(N-2)p-N}{p-1}, \end{aligned}$$

(5.6)

and

$$\begin{aligned} \omega ={\displaystyle \frac{(p+1)q-2p}{p+1} }. \end{aligned}$$

(5.7)

If $q>\frac{2p}{p+1}$ (resp. $q<\frac{2p}{p+1}$), then $\omega >0$ (resp. $\omega <0$) system (5.7) is a perturbation of the Lane–Emden system

$$\begin{aligned} \begin{aligned} x_t&=\frac{2}{p-1}x-y\\ y_t&=-Ky+x^p, \end{aligned} \end{aligned}$$

(5.8)

at $\infty$ (resp. $-\infty$). The following energy type function introduced in [20] is natural with (5.8)

$$\begin{aligned} {\mathcal {N}}(t)={\mathcal {L}}(x(t),y(t))={\displaystyle \frac{K}{p-1} }x^2-{\displaystyle \frac{x^{p+1}}{p+1} } -\left( {\displaystyle \frac{2}{p-1} }\right) ^qMe^{-\omega t}{\displaystyle \frac{x^{q+1}}{q+1} } -{\displaystyle \frac{1}{2} }\left( {\displaystyle \frac{2x}{p-1} }-y\right) ^2, \end{aligned}$$

(5.9)

and it satisfies

$$\begin{aligned} {\mathcal {N}}'(t)&=\left( {\displaystyle \frac{2x}{p-1} }-y\right) \left[ L\left( {\displaystyle \frac{2x}{p-1} }-y\right) -Me^{-\omega t}\left( \left( {\displaystyle \frac{2x}{p-1} }\right) ^q-y^q\right) \right] \nonumber \\&\quad +\omega \left( {\displaystyle \frac{2}{p-1} }\right) ^qMe^{-\omega t}{\displaystyle \frac{x^{q+1}}{q+1} }, \end{aligned}$$

(5.10)

where $L=N-2-\frac{4}{p-1}=K-\frac{2}{p-1}$. Relation (5.10) will be used later on.

(2) For $p,q>1$ set

$$\begin{aligned} u(r)=r^{-\frac{2-q}{q-1}}\xi (t),\quad u'(r)=-r^{-\frac{1}{q-1}}\eta (t),\quad t=\ln r, \end{aligned}$$

(5.11)

then

$$\begin{aligned} \begin{aligned} \xi _t&=\frac{2-q}{q-1}\xi -\eta \\ \eta _t&=-\frac{(N-1)q-N}{q-1}\eta +e^{\overline{\omega } t}\xi ^p+M\eta ^q \end{aligned} \end{aligned}$$

(5.12)

where

$$\begin{aligned} \overline{\omega }={\displaystyle \frac{p-1}{q-1} }\omega . \end{aligned}$$

(5.13)

Note that if $q<\frac{2p}{p+1}$ this system at $\infty$ endows the form

$$\begin{aligned} \begin{aligned} \xi _t&=\frac{2-q}{q-1}\xi -\eta \\ \eta _t&=-\frac{(N-1)q-N}{q-1}\eta +M\eta ^q. \end{aligned} \end{aligned}$$

(5.14)

It is therefore autonomous and much easier to study.

5.1.2 Pohozaev–Pucci–Serrin type functions

Let $\alpha ,\gamma ,\theta ,\kappa$ be real parameters with $\alpha ,\kappa >0$. Set

$$\begin{aligned} {\mathcal {Z}}(r)=r^\kappa \left( {\displaystyle \frac{u'^2}{2} }+{\displaystyle \frac{u^{p+1}}{p+1} }+\alpha {\displaystyle \frac{uu'}{r} }-\gamma u'\left| u'\right| ^q\right) . \end{aligned}$$

(5.15)

This type of function has been introduced in [25] in their study of Eq. (1.2) with $M=1$ with specific parameters. We use it here to embrace all the values of M. We define ${\mathcal {U}}$ by the identity

$$\begin{aligned} {\mathcal {Z}}'+\theta \left| u'\right| ^{q-1}{\mathcal {Z}}=r^{\kappa -1}{\mathcal {U}}. \end{aligned}$$

(5.16)

Then

$$\begin{aligned} {\mathcal {U}}&=\left( \frac{\kappa }{2}+\alpha +1-N\right) u'^2+\left( \frac{\kappa }{p+1}-\alpha \right) u^{p+1}\nonumber \\&\quad +\alpha (\kappa -N){\displaystyle \frac{uu'}{r} }+\left( {\displaystyle \frac{\theta }{p+1} }-\gamma q\right) ru^{p+1}\left| u'\right| ^{q-1}\nonumber \\&\quad +\left( M+\gamma +\frac{\theta }{2}\right) r\left| u'\right| ^{q+1}+\left( \left( (N-1)q-\kappa \right) \gamma \right. \nonumber \\&\quad \left. -\,\alpha (\theta +M)\right) u\left| u'\right| ^{q}-\gamma (\theta +qM)ru\left| u'\right| ^{2q-1}. \end{aligned}$$

(5.17)

5.2 Some known results in the case $M<0$

We recall the results of [14, 25] and [23] relative to the case $M<0$.

Theorem 5.2

• 1. Let $N\ge 3$ and $1<p\le \frac{N}{N-2}$.

  • 1-(i) If $q>\frac{2p}{p+1}$, there is no ground state for any $M<0$ [25, Theorem C].

  • 1-(ii) If $1<q<\frac{2p}{p+1}$ there exists a ground state when |M| is large [14, Proposition 5.7] and there exists no ground state when |M| is small [23].

• 2 Assume $\frac{N}{N-2}<p<\frac{N+2}{N-2}$ and let $\overline{q}$ be the unique root in $(\frac{2p}{p+1},p)$ of the quadratic equation

  $$\begin{aligned} (N-1)(X-p)^2-(N+2-(N-2)p)((p+1)X-2p)X=0. \end{aligned}$$

  • 2-(i) If $\overline{q}\le q<p$ there exists no ground state for any $M<0$ [25, Theorem C].

  • 2-(ii) If $\frac{2p}{p+1}<q<\overline{q}$, there exists no ground state for |M|. It is an open question whether there could exist a finite number of M for which there exists a ground state [25, Theorem 4].

  • 2-(iii) If $1<q<\frac{2p}{p+1}$, there exists a ground state for large |M| [14, Proposition 5.7] and no ground state when |M| is small [23].

• 3 Assume $p>\frac{N+2}{N-2}$ and $q>1$ and let $Q_{N,p}=\frac{2(N-1)p}{2N+p+1}\in (\frac{2p}{p+1},p)$.

  • 3-(i) If $Q_{N,p}<q<p$ there exists a ground state for |M| small.

  • 3-(ii) If $1<q\le Q_{N,p}$ there exists a ground state for any $M<0$ [25, Theorem A].

• 4 Assume $p=\frac{N+2}{N-2}$. There exists at least one $M<0$ such that there exists a ground state if and only if $1<q<p$. More precisely:

  • 4-(i) If $\frac{2p}{p+1}<q<p$ there exists ground state if |M| is small [25, Theorem B].

  • 4-(ii) If $q\ge \frac{2p}{p+1}$ there exists a ground state for any $M<0$ [25, Theorem A].

Remark

It is interesting to quote that when $M<0$ and $q\ge \frac{2p}{p+1}$, there holds [25, Theorem 3],

$$\begin{aligned} u(r)=O(r^{-\frac{2}{p-1}})\quad \text {and }\;u'(r)=O(r^{-\frac{p+1}{p-1}})\quad \text {when }r\rightarrow \infty . \end{aligned}$$

5.3 The case $M>0$

The next result is a consequence of Theorem A.

Theorem 5.3

Let $M>0$, $p>1$ and $q>\frac{2p}{p+1}$ then there exists no radial ground state satisfying $u(0)=1$ when M is large.

Proof

Suppose that such a solution u exists. From Theorem A and Proposition 2.1 there holds

$$\begin{aligned} \sup _{r>0}\left| u'(r)\right| \le c_{N,p,q}|M|^{-\frac{p+1}{(p+1)q-2p}}\quad \text {and }\; \sup _{r>0}r^{\frac{p+1}{p-1}}\left| u'(r)\right| \le c_{N,p}. \end{aligned}$$

(5.18)

As a consequence, if $r>R>0$,

$$\begin{aligned} 1-u(r)= & {} u(0)-u(r)=u(0)-u(R)+u(R)-u(r)\\\le & {} c_{N,p,q}|M|^{-\frac{p+1}{(p+1)q-2p}}R+{\displaystyle \int _{R}^{\infty }}\left| u'(s)\right| ds\\\le & {} c_{N,p,q}|M|^{-\frac{p+1}{(p+1)q-2p}}R+c'_{N,p}R^{-\frac{2}{p-1}}, \end{aligned}$$

with $c'_{N,p}=\frac{p-1}{2}c_{N,p}$. Since $u(r)\rightarrow 0$ when $r\rightarrow \infty$, we take $R=|M|^{\frac{p-1}{(p+1)q-2p}}$ and derive

$$\begin{aligned} 1\le \left( c_{N,p,q}+c'_{N,p}\right) |M|^{-\frac{2}{(p+1)q-2p}}, \end{aligned}$$

(5.19)

and the conclusion follows. $\square$

Remark

If we use Proposition 5.1 we can make estimate (5.19) more precise.

5.3.1 The case $M>0$, $1<p\le \frac{N+2}{N-2}$

It is a consequence of our general results that there is no radial ground state for large M or for small M when $1<q\le \frac{2p}{p+1}$ and $1<p<\frac{N+2}{N-2}$. Indeed, if $1<q<\frac{2p}{p+1}$ is a consequence of the equivalence statement between a priori estimate and non-existence of ground state proved in [23], and if $q= \frac{2p}{p+1}$ it follows from Theorems C and E. Actually in the radial case, the result is more general.

Theorem 5.4

Let $M>0$ and $1< p<\frac{N+2}{N-2}$. If $1<q\le p$, there exists no radial ground state for any M. If $q>p$ there exists no radial ground state for M small enough.

Proof

By Proposition 2.1, we may assume $N\ge 3$ and

$$\begin{aligned} \frac{N}{N-2}<p\le \frac{N+2}{N-2}\quad \text {and }\;q>\frac{N}{N-1}. \end{aligned}$$

(5.20)

(i) Assume first $q< \frac{2p}{p+1}$. We use the system (5.5). Then $\omega$, defined by (5.7) is negative. Hence the Leighton function ${\mathcal {N}}$ defined by (5.9) is nonincreasing since $L\le 0$ when $p\le \frac{N+2}{N-2}$. Furthermore since $(x(t),y(t))\rightarrow (0,0)$ when $t\rightarrow -\infty$ and $e^{-\omega t}\rightarrow 0$, we get ${\mathcal {N}}(-\infty )=0$ it follows that ${\mathcal {N}}(t)<0$ for $t\in \mathbb {R}$. Moreover, by Proposition 2.1,

$$\begin{aligned} u(r)=O(r^{-\frac{2-q}{q-1}})\quad \text {as }r\rightarrow \infty \Longleftrightarrow x(t)=O(e^{\frac{q(p+1)-2p}{(p-1)(q-1)}t})=o(1)\quad \text {as }t\rightarrow \infty \end{aligned}$$

This implies $e^{-\omega t}x^{q+1}(t)=O(e^{2\frac{q(p+1)-2p}{(p-1)(q-1)}t})=o(1)$ as $t\rightarrow \infty$ and ${\mathcal {N}}(\infty )=0$, contradiction.

(ii) Assume next $\frac{2p}{p+1}\le q\le p$. We consider the function (5.15) with the parameters

$$\begin{aligned} \kappa ={\displaystyle \frac{2(p+1)(N-1)}{p+3} }=(p+1)\alpha \quad \text {and }\;\gamma =-{\displaystyle \frac{2M}{q(p+1)+2} }={\displaystyle \frac{\theta }{q(p+1)} }, \end{aligned}$$

already used by [25] when $M=-1$, and we get with ${\mathcal {U}}$ defined by (5.16),

$$\begin{aligned} {\mathcal {U}}={\displaystyle \frac{2}{(p+3)^2} }{\displaystyle \frac{u\left| u'\right| }{r} }\left( A+BM\chi +CM\chi ^2\right) \quad \text {with}\;\chi ={\displaystyle \frac{p+3}{2+q(p+1)} }r\left| u'\right| ^{q-1}, \end{aligned}$$

where

$$\begin{aligned} A=(N-1)(N+2-(N-2)p)\,,\; B=2(N-1)(p-q)\,,\; C=q(q(p+1)-2p). \end{aligned}$$

(5.21)

By our assumptions $A\ge 0$, $B\ge 0$ and $C>0$. Hence ${\mathcal {U}}>0$. This implies

$$\begin{aligned} {\mathcal {Z}}(r)= & {} e^{-\int _{0}^{r}\theta \left| u'\right| ^{q-1}ds}{\mathcal {Z}}(0)+{\displaystyle \int _{0}^{r}} e^{-\theta \int _{s}^{r}\left| u'\right| ^{q-1}d\sigma }s^{\kappa -1}{\mathcal {U}}(s)ds\\= & {} {\displaystyle \int _{0}^{r}}e^{-\theta \int _{s}^{r}\left| u'\right| ^{q-1}d\sigma }s^{\kappa -1}{\mathcal {U}}(s)ds, \end{aligned}$$

since ${\mathcal {Z}}(0)=0$. If u is a ground state, then $u'(r)\rightarrow 0$ as $r\rightarrow \infty$, thus $u\left| u'\right| ^q\le u\left| u'\right| ^{\frac{2p}{p+1}}$. Hence, from Proposition 2.1, $u'^2(r)=O(r^{-2\frac{p+1}{p-1}})$ as $r\rightarrow \infty$. The other terms $u^{p+1}(r)$, $r^{-1}u(r)u'(r)$ and $u\left| u'\right| ^{\frac{2p}{p+1}}$ satisfy the same bound, hence

$$\begin{aligned} {\mathcal {Z}}(r)=O(r^{\kappa -\frac{2(p+1)}{p-1}})=O(r^{\frac{2(p+3)(N-1)}{p+3} -\frac{2(p+1)}{p-1}})=O(r^{\frac{2(p+1)((N-2)p-(N+2))}{(p+3)(p-1)}}). \end{aligned}$$

Then ${\mathcal {Z}}(r)\rightarrow 0$ when $r\rightarrow \infty$, contradiction.

(iii) Suppose $q>p$ and u is a ground state. By Proposition 5.1 and (5.18), there holds

$$\begin{aligned} r\left| u'\right| ^{q-1}=r\left| u'\right| ^{\frac{p-1}{p+1}}\left| u'\right| ^{q-\frac{2p}{p+1}}\le c_{N,p}. \end{aligned}$$

Then $\chi =\frac{p+3}{2+q(p+1)}r\left| u'\right| ^{q-1}\le c_{N,p}$. Hence, if $M\le M_{N,p}$ for some $M_{N,p}>0$, ${\mathcal {U}}$ is positive as A is. We conclude as above. $\square$

5.3.2 The case $M>0$, $p>\frac{N+2}{N-2}$

We recall that in Theorem C if $q=\frac{2p}{p+1}$ and $p>1$ there is no ground state whenever $M>M_{N,p}$, see (1.26). In Theorem A$'$ if $1<q<\frac{2p}{p+1}$ and $p>1$ there is no ground state u such that $u(0)=1$ if M is too large. In the next result we complement Theorem 5.3 for small value of M in assuming $q>\frac{2p}{p+1}$.

Theorem 5.5

If $p>\frac{N+2}{N-2}$ and $q\ge \frac{2p}{p+1}$ then there exist radial ground states for $M>0$ small enough.

Proof

First we consider the function ${\mathcal {Z}}$ with $k=N$ and obtain

$$\begin{aligned} {\mathcal {Z}}(r)=r^{N}\left( {\displaystyle \frac{u'^2}{2} }+{\displaystyle \frac{u^{p+1}}{p+1} }+\alpha {\displaystyle \frac{uu'}{r} }-\gamma u\left| u'\right| ^q\right) . \end{aligned}$$

The function vanishes at the origin. We compute ${\mathcal {U}}$ from the identity ${\mathcal {Z}}'+\theta \left| u'\right| ^{q-1}{\mathcal {Z}}=r^{N-1}{\mathcal {U}}$ and get

$$\begin{aligned} {\mathcal {U}}= & {} \left( \alpha -{\displaystyle \frac{N-2}{2} }\right) u'^2+\left( {\displaystyle \frac{N}{p+1} }-\alpha \right) u^{p+1} +\left( {\displaystyle \frac{\theta }{p+1} }-\gamma q\right) ru^{p+1}\left| u'\right| ^{q-1}\\&+\left( M+\gamma +{\displaystyle \frac{\theta }{2} }\right) r\left| u'\right| ^{q+1}+\left[ \left( (N-1)q-N\right) \gamma -\alpha (\theta +M)\right] u\left| u'\right| ^{q}\\&-\gamma (\theta +qM)ru\left| u'\right| ^{2q-1}. \end{aligned}$$

If $\gamma =0$ and $\theta =-2M$, then

$$\begin{aligned} {\mathcal {U}}=\left( \alpha -{\displaystyle \frac{N-2}{2} }\right) u'^2+\left( {\displaystyle \frac{N}{p+1} }-\alpha \right) u^{p+1}-{\displaystyle \frac{2M}{p+1} }ru^{p+1}\left| u'\right| ^{q-1} +\alpha Mu\left| u'\right| ^{q}. \end{aligned}$$

If u is a regular solution which vanishes at some $r_0>0$, then ${\mathcal {Z}}(r_0)=2^{-1}r_0^2u'^N(r_0)>0$. As $p>\frac{N+2}{N-2}$, by choosing $\alpha =\frac{1}{2}\left( \frac{N}{p+1}+\frac{N-2}{2}\right)$ we have $\frac{N}{p+1}<\alpha <\frac{N-2}{2}$. We define $\ell >0$ by $(N-2)p-(N+2)=4(p+1)\ell$, then $\frac{N-2}{2}-\alpha =\alpha -\frac{N}{p+1}=\ell$ and then

$$\begin{aligned} {\mathcal {U}}\le -\ell (u'^2+u^{p+1})+M\alpha u\left| u'\right| ^q. \end{aligned}$$

Assume first $q<2$, we have from Hölder’s inequality and $0<r\le r_0$ where u is positive

$$\begin{aligned} u\left| u'\right| ^q\le {\displaystyle \frac{q}{2} }u'^2+{\displaystyle \frac{2-q}{2} }\left| u\right| ^{\frac{2}{2-q}}\le u'^2+\left| u\right| ^{\frac{2}{2-q}}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {U}}+(\ell -M)u'^2\le & {} M\alpha u^{\frac{2}{2-q}}-\ell u^{p+1}=\ell u^{p+1}\left( {\displaystyle \frac{M\alpha }{\ell } }u^{\frac{q(p+1)-2p}{2-q}}-1\right) \\\le & {} \ell u^{p+1}\left( {\displaystyle \frac{M\alpha }{\ell } }-1\right) \end{aligned}$$

since $q\ge \frac{2p}{p+1}$ and $u\le u(0)=1$. Taking $M\le \frac{\ell }{\alpha }=\frac{(N-2)p-N-2}{(N-2)p+3N-2}$, ${\mathcal {U}}$ is negative implying that $r\mapsto e^{-2M\int _0^r|u'|^{q-1}ds}{\mathcal {Z}}(r)$ is nonincreasing. Since ${\mathcal {Z}}(0)=0$, ${\mathcal {Z}}(r)\le 0$, a contradiction.

If $q=2$, then ${\mathcal {U}}\le -\ell (u'^2+u^{p+1})+M\alpha u'^2$ since $u\le 1$ on $[0,r_0]$. We still infer that ${\mathcal {U}}\le 0$ if we choose $M\le \frac{\ell }{\alpha }$.

Finally, if $q>2$, we have from Theorem A, $u'\le C_{N,p,q} M^{-\frac{p+1}{(p+1)q-2p}}$. Therefore, using again the decay of u from $u(0)=1$,

$$\begin{aligned} M\alpha u\left| u'\right| ^q\le M\alpha u\left| u'\right| ^{q-2}u'^2\le M\alpha C^{q-2}_{N,p,q} M^{-\frac{(p+1)(q-2)}{(p+1)q-2p}}u'^2= \alpha C^{q-2}_{N,p,q}M^{\frac{2}{(p+1)q-2p}}u'^2. \end{aligned}$$

Hence ${\mathcal {U}}\le -\left( \ell -\alpha C^{q-2}_{N,p,q}M^{\frac{2}{(p+1)q-2p}}\right) u'^2$. Taking

$$\begin{aligned} M^{\frac{2}{(p+1)q-2p}}\le C^{2-q}_{N,p,q}\frac{(N-2)p-N-2}{(N-2)p+3N-2} \end{aligned}$$

we conclude that ${\mathcal {U}}<0$ which ends the proof as in the previous cases. $\square$

Theorem F is the combination of Theorems 5.3, 5.4 and 5.5.

6 Separable solutions

We denote by $(r,\sigma )\in \mathbb {R}_+\times S^{N-1}$ the spherical coordinates in $\mathbb {R}^N$. Then Eq. (1.2) takes the form

$$\begin{aligned} - u_{rr} -{\displaystyle \frac{N-1}{r} } u_{r} -{\displaystyle \frac{1}{r^2} }\Delta 'u=|u|^{p-1}+M\left( u_{r} ^2+{\displaystyle \frac{1}{r^2} }\left| \nabla 'u\right| ^2\right) ^{\frac{q}{2}}, \end{aligned}$$

(6.1)

where $\Delta '$ is the Laplace–Beltrami operator on $S^{N-1}$ and $\nabla '$ the tangential gradient. If we look for separable nonnegative solutions of (1.2) i.e. solutions under the form $u(r,\sigma )=\psi (r)\omega (\sigma )$, then $q=\frac{2p}{p+1}$, $\psi (r)=r^{-\frac{2}{p-1}}$, and $\omega$ is a solution of

$$\begin{aligned} -\Delta '\omega +{\displaystyle \frac{2K}{p-1} }\omega =\omega ^p+M\left( \left( {\displaystyle \frac{2}{p-1} } \right) ^2\omega ^2+\left| \nabla '\omega \right| ^2\right) ^{\frac{p}{p+1}}, \end{aligned}$$

(6.2)

where K is defined in (5.6). Throughout this section we assume

$$\begin{aligned} p>1\quad \text {and}\quad q=\frac{2p}{p+1}. \end{aligned}$$

(6.3)

6.1 Constant solutions

The constant function $\omega =X$ is a solution of (6.2) if

$$\begin{aligned} X^{p-1}+M\left( {\displaystyle \frac{2}{p-1} }\right) ^{\frac{2p}{p+1}}X^{\frac{p-1}{p+1}} -{\displaystyle \frac{2K}{p-1} }=0. \end{aligned}$$

(6.4)

For $N=1, 2$ and $p>1$ or $N\ge 3$ and $1<p<\frac{N}{N-2}$, we recall that $\mu ^*=\mu ^*(N)$ has been defined in (1.24). The following result is easy to prove

Proposition 6.1

• 1. Let $M\ge 0$ then there exists a unique positive root $X_M$ to (6.4) if and only if $p>\frac{N}{N-2}$. Moreover the mapping $M\mapsto X_M$ is continuous and decreasing from $[0,\infty )$ onto $(0,\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}]$.

• 2. Let $M<0$, $N\ge 3$ and $p\ge \frac{N}{N-2}$ then there exists a unique positive root $X_M$ to (6.4) and the mapping $M\mapsto X_M$ is continuous and decreasing from $(-\infty ,0]$ onto $[\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}},\infty )$.

• 3. Let $M<0$, $N=1,2$ and $p>1$ or $N\ge 3$ and $1<p<\frac{N}{N-2}$ then there exists no positive root to (6.4) if $-\mu ^*<M\le 0$. If $M=M^*:=-\mu ^*$ there exists a unique positive root $X_{M^*}=\left( \frac{2|K|}{p(p-1)}\right) ^{\frac{1}{p-1}}$. If $M<-\mu ^*$ there exist two positive roots $X_{1,M}<X_{2,M}$. The mapping $M\mapsto X_{1,M}$ is continuous and increasing from $(-\infty ,\mu ^*]$ onto $(0,X_{M^*}]$. The mapping $M\mapsto X_{2,M}$ is continuous and decreasing from $(-\infty ,\mu ^*]$ onto $[X_{M^*},\infty )$.

Abridged proof

Set

$$\begin{aligned} f_M(X)=X^{p-1}+M\left( {\displaystyle \frac{2}{p-1} }\right) ^{\frac{2p}{p+1}}X^{\frac{p-1}{p+1}} -{\displaystyle \frac{2K}{p-1} }, \end{aligned}$$

(6.5)

then $f'_M(X)=(p-1)X^{p-2}+M\frac{p-1}{p+1}\left( {\displaystyle \frac{2}{p-1} } \right) ^{\frac{2p}{p+1}}X^{-\frac{2}{p+1}}$.

• 1. If M is nonnegative, $f_M$ is increasing from $-\frac{2K}{p-1}=-\frac{2[(N-2)p-N]}{(p-1)^2}$ to $\infty$; hence, if $p>\frac{N}{N-2}$ there exists a unique $X_M> 0$ such that $f_M(X_M)=0$, while if $1<p<\frac{N}{N-2}$, $f_M$ admits no zero on $[0,\infty )$. Since $f_M>f_{M'}$ for $M>M'>0$, there holds $X_M>X_{M'}$, By the implicit function theorem the mapping $M\mapsto X_M$ is $C^1$ and decreasing from $[0,\infty )$ onto $(0,\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}]$. Actually it can be proved that (see [10, Proposition 2.2])

  $$\begin{aligned} X_M=\frac{p-1}{2}\left( \frac{K}{M}\right) ^{\frac{p+1}{p-1}}(1+o(1))\quad \text {as }\;M\rightarrow \infty . \end{aligned}$$

  (6.6)

• 2. If M is negative, $f_M$ achieves it minimum on $[0,\infty )$ at $X_0=\left( \frac{-M}{p+1}\right) ^\frac{p+1}{p(p-1)}\left( \frac{2}{p-1}\right) ^\frac{2}{p-1}$, and

  $$\begin{aligned} f_M(X_0)= & {} -{\displaystyle \frac{p}{\left( p+1\right) ^{\frac{p+1}{p}}} }\left( {\displaystyle \frac{2}{p-1} }\right) ^{2}(-M)^{\frac{p+1}{p}}-{\displaystyle \frac{2K}{p-1} }\\= & {} -\left( {\displaystyle \frac{2}{p-1} }\right) ^{2}\left( {\displaystyle \frac{p}{\left( p+1\right) ^{\frac{p+1}{p}}} }(-M)^{\frac{p+1}{p}}+{\displaystyle \frac{(N-2)p-N}{2} }\right) . \end{aligned}$$

  Since $K>0$, there exists a unique $X_M>0$ such that $f_M(X_M)=0$ and $X_M>X_0$. The mapping $M\mapsto X_M$ is $C^1$ and decreasing from $(-\infty , 0]$ onto $[\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}},\infty )$. The following estimate holds

  $$\begin{aligned}&\max \left\{ \left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}, \left( \frac{2}{p-1}\right) ^{\frac{2}{p-1}}|M|^{\frac{p+1}{p(p-1)}}\right\} \le X_M\nonumber \\&\quad \le 2^{\frac{2}{p-1}}\left( \left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}} +\left( \frac{2}{p-1}\right) ^{\frac{2}{p-1}}|M|^{\frac{p+1}{p(p-1)}}\right) . \end{aligned}$$

  (6.7)

• 3. If $N=1,2$ and $p>1$ or $N\ge 3$ and $1<p<\frac{N}{N-2}$, then $f_M(0)>0$. Hence, if $f_M(X_0)>0$ there exists no positive root to $f_M(X)=0$. Equivalently, if $-\mu ^*<M<0$. If $f_M(X_0)=0$, $X_0$ is a double root and this is possible only if $M=-\mu ^*$, hence $X_{-\mu ^*}=\left( \frac{2|K|}{p(p-1)}\right) ^{\frac{1}{p-1}}$. If $f_M(X)<0$, or equivalently, if $M<-\mu ^*$, the equation $f_M(X)=0$ admits two positive roots $X_{1,M}<X_0<X_{2,M}$. The monotonicity of the $X_{j,M}$, j=1,2, and their range follows easily from the monotonicity of $M\mapsto f_M(X)$ for $M<0$. Actually the following asymptotics hold when $M\rightarrow -\infty$,

  $$\begin{aligned} X_{1,M}=\frac{p-1}{2}\left( \frac{K}{M}\right) ^{\frac{p+1}{p-1}}(1+o(1))\; \text {and }\;\, X_{2,M}=\left( \frac{2}{p-1}\right) ^\frac{2}{p-1}\left( -M\right) ^{\frac{p+1}{p(p-1)}} (1+o(1)). \end{aligned}$$

  (6.8)

  $\square$

6.2 Bifurcations

We set

$$\begin{aligned} A(\omega )= -\Delta '\omega +{\displaystyle \frac{2K}{p-1} }\omega -\omega ^p-M\left( \left( {\displaystyle \frac{2}{p-1} }\right) ^2\omega ^2+\left| \nabla '\omega \right| ^2\right) ^{\frac{p}{p+1}}, \end{aligned}$$

(6.9)

If $\eta \in C^\infty (S^{N-1})$ and if there exists a constant positive solution X to $A(X)=0$ we have

$$\begin{aligned} {\displaystyle \frac{d}{d\tau } }A(X+\tau \eta )\lfloor _{\tau =0}\,=\,-\Delta '\eta +\left( {\displaystyle \frac{2K}{p-1} } -pX^{p-1}-M{\displaystyle \frac{2p}{p+1} }\left( {\displaystyle \frac{2}{p-1} }\right) ^{\frac{2p}{p+1}} X^{\frac{p-1}{p+1}}\right) \eta . \end{aligned}$$

Hence the problem is singular if

$$\begin{aligned} -{\displaystyle \frac{2K}{p-1} }+pX^{p-1}+M{\displaystyle \frac{2p}{p+1} }\left( {\displaystyle \frac{2}{p-1} } \right) ^{\frac{2p}{p+1}}X^{\frac{p-1}{p+1}}=\lambda _k, \end{aligned}$$

(6.10)

where $\lambda _k=k(k+N-2)$ is the k-th nonzero eigenvalue of $-\Delta '$ in $H^1(S^{N-1})$. The following result follows classically from the standard bifurcation theorem from a simple eigenvalue (which can always be assumed if we consider functions depending only on the azimuthal angle on $S^{N-1}$ reducing the eigenvalue problem to a simple Legendre type ordinary differential equation) see e.g. [26, Chapter 13] and identity (6.4).

Theorem 6.2

Let $M_0\in \mathbb {R}$ and $X_{M_0}$ be a constant solution of (6.2). If $X_{M_0}$ satisfies for some $k\in \mathbb {N}^*$,

$$\begin{aligned} {M_0}\left( {\displaystyle \frac{2}{p-1} }\right) ^{\frac{2p}{p+1}}X_{M_0}^{\frac{p-1}{p+1}}={\displaystyle \frac{p+1}{p(p-1)} }\left( 2K-\lambda _k\right) , \end{aligned}$$

(6.11)

there exists a continuous branch of nonconstant positive solutions $(M,\omega _M)$ of (6.2) bifurcating from the $({M_0},X_{M_0})$.

Since $M\left( {\displaystyle \frac{2}{p-1} }\right) ^{\frac{2p}{p+1}}X_M^{\frac{p-1}{p+1}}={\displaystyle \frac{2K}{p-1} }-X_M^{p-1}$ by (6.4) the following statements follow immediately from Proposition 6.1.

Lemma 6.3

Set $\Phi (M)=M\left( \frac{2}{p-1}\right) ^{\frac{2p}{p+1}}X_M^{\frac{p-1}{p+1}}$ when $X_M$ is uniquely determined, and $\Phi _j(M)=M\left( \frac{2}{p-1}\right) ^{\frac{2p}{p+1}}X_{j,M}^{\frac{p-1}{p+1}}$, j=1,2, if there exist two equilibria. Then

• 1. If $N\ge 3$ and $p>\frac{N}{N-2}$, the mapping $M\mapsto \Phi (M)$ is continuous and increasing from $[0,\infty )$ onto $[0,\frac{2K}{p-1})$.

• 2. If $N\ge 3$ and $p\ge \frac{N}{N-2}$, the mapping $M\mapsto \Phi (M)$ is continuous and increasing from $(-\infty ,0]$ onto $(-\infty ,0]$.

• 3. If $N=1,2$ and $p>1$ or $N\ge 3$ and $1<p<\frac{N}{N-2}$, the mapping $M\mapsto \Phi _1(M)$ (resp $M\mapsto \Phi _2(M))$ is continuous and decreasing (resp. increasing) from $(-\infty ,-\mu ^*]$ onto $[\frac{2K}{p-1},0)$ (resp. $(-\infty ,\frac{2K}{p-1}])$.

If we analyse the range $R[\Phi ]$ of $\Phi$ or $R[\Phi _j]$ of $\Phi _j$, we prove the following result.

Theorem 6.4

• 1 Let $N\ge 3$ and $p\ge \frac{N}{N-2}$.

  • 1-(i) There exists a continuous curve of bifurcation $(M,\omega _M)$ issued from $(M_0,X_{M_0})$ for some $M_0=M_0(p)\ge 0$ if and only if $p\ge \frac{N+1}{N-3}$ and $k=1$. Furthermore $M_0(\frac{N+1}{N-3})=0$.

  • 1-(ii) The bifurcation curve $s\mapsto (M(s),\omega _{M(s)})$, is defined on $(-\epsilon _0,\epsilon _0)$ for some $\epsilon _0>0$ and verifies $(M(0),\omega _{M(0)})=(M_0,X_{M_0})$.

• 2. Let $N\ge 3$ and $p\ge \frac{N}{N-2}$.

  • 2-(i) For any $k\ge 1$ there exist $M_k<0$ and a continuous branch of bifurcation $(M,\omega _M)$ issued from $(M_k,X_{M_k})$, with the restriction that $p< \frac{N+1}{N-3}$ if $k=1$.

  • 2-(ii) The bifurcation curve $s\mapsto (M(s),\omega _{M(s)})$, is defined on $(-\epsilon _0,\epsilon _0)$ for some $\epsilon _0>0$ and verifies $(M(0),\omega _{M(0)})=(M_0,X_{M_0})$. Finally $M_k\rightarrow -\infty$ when $k\rightarrow \infty$.

• 3. let $N=1,2$ and $p>1$, or $N\ge 3$ and $1<p<\frac{N}{N-2}$.

  • 3-(i) There exists no $M<0$ such that $\frac{2K}{p-1}<\Phi _1 (M)<0$, and a countable set of $M_k<0$, $k\ge 1$, such that $\Phi _2(M_k)=\frac{p+1}{p(p-1)}\left( 2K-\lambda _k\right)$.

  • 3-(ii) There exist a countable branches of bifurcation of solutions $(M_k(s),\omega _{M_k(s)})$ issued from $(M_k, X_{2,M_k})$.

Proof

Assertion 1. Since from Lemma 6.3, $R[\Phi ]=[0,\frac{2K}{p-1})$ for $M\ge 0$, we have to see under what condition on $p\ge \frac{N}{N-2}$ one can find $k\ge 1$ such that

$$\begin{aligned} 0\le {\displaystyle \frac{p+1}{p(p-1)} }\left( 2K-\lambda _k\right)<\frac{2K}{p-1}\Longleftrightarrow \frac{2K}{p+1} <\lambda _k\le 2K. \end{aligned}$$

Since $K<N$ and $\lambda _k\ge 2N$ for $k\ge 2$, the only possibility for this last inequality to hold is $k=1$. The inequality $\frac{2K}{p+1} <N-1$ always holds since $p>1$, while the inequality $N-1=\lambda _1\le 2K$ is equivalent to $p\ge \frac{N+1}{N-3}$. Therefore $M_0=0$ and $X_{M_0}=\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}$. If we consider only functions on the sphere $S^{N-1}$ which depend uniquely on the azimuthal angle $\theta =\tan ^{-1}(x_N\lfloor _{S^{N-1}})$, the function $\psi _1(\sigma )= x_N\lfloor _{S^{N-1}}$ is a eigenfunction of $-\Delta '$ in $H^1(S^{N-1})$ with multiplicity one. Hence the bifurcation branch is locally a regular curve $s\mapsto (M(s),\omega _{M(s)})$ with $0\le s<\epsilon '_0$ and we construct the bifurcating solution on $S^{N-1}$ using the classical tangency condition [26, Theorem 13.5],

$$\begin{aligned} \omega _{M(s)}=X_{M_0}+s(\psi _1+\zeta _s) \end{aligned}$$

(6.12)

where $\zeta _s\in H^{1}(S^{N-1})$, is orthogonal to $\psi _1$ in $H^{1}(S^{N-1})$ and satisfies $\left\| \zeta _s\right\| _{C^1}=o(1)$ when $s\rightarrow 0$. This implies the claim.

Assertion 2. Since $R[\Phi ]=(-\infty ,0)$ for $M< 0$, we have to find $k\ge 1$ such that

$$\begin{aligned} {\displaystyle \frac{p+1}{p(p-1)} }\left( 2K-\lambda _k\right)< 0\Longleftrightarrow 2K< \lambda _k. \end{aligned}$$

As in Case 1, $K<2N$, then inequality $2K\le \lambda _k$ holds for all $k\ge 2$, and if $k=1$ this is possible only if $p< \frac{N+1}{N-3}$. The construction of the bifurcating curve is the same as in Case 1.

Assertion 3. We have $R[\Phi _1]=[\frac{2K}{p-1},0)$ for $M\le -\mu ^*$. If we look for the existence of some $k\ge 1$ such that

$$\begin{aligned} \frac{2K}{p-1}\le {\displaystyle \frac{p+1}{p(p-1)} }\left( 2K-\lambda _k\right)<0\Longleftrightarrow 2K\le \lambda _k<\frac{2K}{p+1}; \end{aligned}$$

we get an impossibility since $K<0$. Hence there exists no $M_0<0$ such that $(M_0,X_{1,M_0})$ is a bifurcation point. We have also $R[\Phi _2]=(-\infty ,\frac{2K}{p-1}]$ for $M\le -\mu ^*$. Now the condition for the existence of a bifurcation branch issued from $(M_0,X_{2,M_0})$ for some $M_0\le -\mu ^*$ is

$$\begin{aligned} {\displaystyle \frac{p+1}{p(p-1)} }\left( 2K-\lambda _k\right) \le \frac{2K}{p-1}\Longleftrightarrow \lambda _k\ge \frac{2K}{p+1}, \end{aligned}$$

which is always true for any $k\ge 1$ and $1<p<\frac{N}{N-2}$. $\square$

Remark

The exponent $p=\frac{N+1}{N-3}$ is the Sobolev critical exponent on $S^{N-1}$. If one consider the Lane–Emden equation with a Leray potential

$$\begin{aligned} -\Delta u+\lambda |x|^{-2}u=u^{\frac{N+1}{N-3}}, \end{aligned}$$

(6.13)

with $\lambda \in \mathbb {R}$, then the separable solutions $u(r,\sigma )=r^{-\frac{N-3}{2}}\omega (\sigma )$ verify

$$\begin{aligned} -\Delta '\omega +\left( \frac{(N-1)(N-3)}{4}-\lambda \right) \omega -\omega ^{\frac{N+1}{N-3}}=0\quad \text {on }\; S^{N-1}. \end{aligned}$$

(6.14)

It was observed in [7] that there exists a branch of bifurcation $(\lambda ,\omega _\lambda )$ with $\lambda >0$ issued from $(0,\omega _0)$, where $\omega _0$ is the constant explicit solution of (6.14).

Remark

In Theorem 6.4-1 and the above remark, we conjectured that on the bifurcating curve there holds locally $M(s)<M_0$, and that for any $p\ge \frac{N+1}{N-3}$ there exists $M_0:=M_0(p)$ such that for $M> M_0$ all the positive solutions to (6.2) are constant, furthermore $M_0$ is defined by (6.11). When $p=\frac{N+1}{N-3}$, then $M=0$ and there exists infinitely many positive solutions to (6.2) [7, Proposition 5.1]. When $\frac{N}{N-2}<p< \frac{N+1}{N-3}$, it is unclear if the branches of bifurcation $(M(s),\omega _{M(s)})$ issued from $(M_0,\omega _{M_0})$ with $M_0<0$ are such that M(s) keeps a constant sign. If it is the case one could expect that if $M\ge 0$ and $\frac{N}{N-2}<p< \frac{N+1}{N-3}$, all the positive solutions to (6.2) are constant.

The following statement is an immediate consequence of Theorem 6.4.

Corollary 6.5

• 1. If $p>1$ and $q=\frac{2p}{p+1}$ there always exist nonradial positive singular solutions of (1.2) in $\mathbb {R}^N{\setminus }\{0\}$ under the form $u(r,\sigma )=r^{-\frac{2}{p-1}}\omega (\sigma )$.

• 2. If $N\ge 4$ and $p>\frac{N+1}{N-3}$, the functions are obtained by bifurcation from $X_M$ with $M>0$.

• 3. If $N\ge 3$ and $\frac{N}{N-2}\le p<\frac{N+1}{N-3}$, the functions are obtained by bifurcation from $X_M$ with $M<0$.

• 4. If $N=1,2$ and $p>1$ or $N\ge 3$ and $1<p<\frac{N}{N-2}$, the functions are obtained by bifurcation from $(M_k, X_{2,M_k})$ with $M_k<-\mu ^*$ and $k\ge 1$.