On a problem of Huang concerning best constants in Sobolev embeddings

Abstract

Answering a question raised by Y.X. Huang, we prove what follows: if $\Omega$ is a bounded smooth domain and $p>1$, then the mapping $q\mapsto \lambda _q|\Omega |^\frac{p}{q}$ is decreasing in $]0,p^*[$ and Lipschitz continuous on compact subsets of $]0,p^*[, \lambda _q$ being the $p$-th power of the best Sobolev constant for the embedding of $W^{1,p}_0(\Omega )$ into $L^q(\Omega )$.

1 Introduction and main result

The present paper is devoted to the classical Dirichlet problem for the $p$-Laplacian operator

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _p u=\lambda |u|^{q-2}u &{}\text{ in } \Omega \\ u=0 &{}\text{ on } \partial \Omega \end{array},\right. \end{aligned}$$

(1)

where $\Omega \subset {\mathbb R}^N$ ($N>1$) is a bounded domain with a boundary $\partial \Omega$ of class $C^{1,\alpha }$ ($0<\alpha \le 1$), $p>1, 0<q<p^*$ are real numbers (we recall that the Sobolev critical exponent is $p^*=Np/(N-p)$ if $1<p<N, p^*=+\infty$ if $p\ge N$), and $\lambda >0$ is a parameter.

Problem (1) has been widely investigated, with different results according to the relation between the exponents $p$ and $q$. For the homogenous case ($p=q$), we refer the reader to Lê [13] for a detailed description of the eigenvalues and eigenspaces. Ôtani [16] proved that, for $\lambda =1$, problem (1) admits at least one nonnegative solution for $1<q<p^*$, and, if $\Omega$ is strictly star-shaped, it admits no non-negative, nonzero solutions for $q=p^*$ and no nontrivial solutions for $q>p^*$. Uniqueness of the positive solution with minimal energy for $1<q<p, \lambda =1$ was proved by Franzina and Lamberti [8], who reduced problem (1) to a homogeneous one by means of a nonlocal term of the type $\Vert u\Vert _p^{q-p}$. Most of the mentioned results are based on variational methods, and they all deal with the case $1<q<p^*$.

For $0<q<1$, problem (1) is singular at $0$ and it cannot be directly studied in a variational framework. For the semilinear case ($p=2$), existence results for related singular equations were proved by Crandall et al. [3] and by Lazer and McKenna [12], while uniqueness of the solution was examined by Diaz et al. [4]. A bifurcation result for a semilinear equation with a singular perturbation was obtained by Cîrstea et al. [2]. In most cases, the study of singular problems is based on sub- and super-solutions.

Problem (1) is strictly related to the best constant in the Sobolev embeddings. Let us consider the Sobolev space $W^{1,p}_0(\Omega )$ and the Lebesgue space $L^q(\Omega )$, endowed with the norms $\Vert \nabla u\Vert _p, \Vert u\Vert _q$, respectively. By the Sobolev theorem, we have

$$\begin{aligned} \inf _{u\in W^{1,p}_0(\Omega ), \ u\ne 0}\frac{\Vert \nabla u\Vert _p^p}{\Vert u\Vert _q^p}=\lambda _q\in ]0,+\infty [. \end{aligned}$$

(2)

The constant $\lambda _q$ was explicitly determined by Talenti [19] for $\Omega ={\mathbb R}^N, q=p^*$, but it is not known in general. It can be proved that for $\lambda =\lambda _q$, problem (1) has a positive smooth solution $u_q$ s.t. $\Vert u_q\Vert _q=1$ and $\Vert \nabla u_q\Vert _p=\lambda _q^\frac{1}{p}$. In his interesting paper [11], Huang proved that the mapping $q\mapsto \lambda _q$ is continuous in $]1,p[$ and upper semicontinuous in $]p,p^*[$. Later, he put forward the following conjecture: We suspect that $\lambda _q$ has some monotonicity with respect to $q$. In fact, one can prove that, if $|\Omega |\le 1$, then $\lambda _q\le \lambda _r$ if $r<q$ (here and in the sequel, $|\Omega |$ denotes the Lebesgue $N$-dimensional measure of $\Omega$).

The aim of the present note is to give such question an answer, which turns out to be positive (actually, we will prove much more). Our main result is the following:

Theorem 1

The mapping $g:]0,p^*[\rightarrow ]0,+\infty [$, defined by

$$\begin{aligned} g(q)=\lambda _q|\Omega |^{\frac{p}{q}} \quad \text{ for } \text{ all }\quad q\in ]0,p^{*}{[}, \end{aligned}$$

is Lipschitz continuous on compact subsets of $]0,p^*[$ and decreasing in $]0,p^*[$.

Our result both improves that of Huang and answers the question about monotonicity. Indeed, clearly from Theorem 1 it follows that $q\mapsto \lambda _q$ is continuous on the whole interval $]0,p^*[$. Moreover, if $|\Omega |\le 1$, then for all $0<r<q<p^*$ we have from $g(r)>g(q)$ that

$$\begin{aligned} \lambda _q<\lambda _r|\Omega |^\frac{p(q-r)}{rq}\le \lambda _r. \end{aligned}$$

For a surprising coincidence we became aware, after writing the present paper, that a partial positive answer to the problem raised by Huang had been given by Ercole in his very recent paper [6]. The author deals with properties of the map $q\mapsto \lambda _q$ when $p<N$ and proves that it is absolutely continuous in $[1,p^*]$. Our result is more general as in our arguments $p$ is any real number in $]1,+\infty [$ and the properties of $\lambda _q$ are also studied in $]0,1]$ (for more details see Remark 10).

The proof of Theorem 1 is delivered as follows (see Sect. 3). First, by applying Hölder inequality, we prove that $g$ is Lipschitz continuous on compact subsets of $]0,p^*[$ and nonincreasing in $]0,p^*[$. Then, dealing with the more delicate issue of strict monotonicity, we split our study in two parts: for $0<q\le 1$, by means of sub- and super-solutions, we prove existence and some estimates for a positive solution of class $C^1(\Omega )$ of the singular problem (1), which in turn imply that $g$ is decreasing in $]0,1]$; for $1<q<p^*$, by variational methods, we prove the existence of a positive solution with higher regularity $C^1(\overline{\Omega })$, from which, by topological arguments, we deduce that $g$ is decreasing in $]1,p^*[$.

A possible application of Theorem 1 is toward the study of the asymptotic behavior of the pair $(\lambda _q,u_q)$, which has been addressed by many authors (see for instance Lee [15] and Garcia Azorero and Peral Alonso [9]). Namely we will prove (see Sect. 4) that $\lambda _q\rightarrow \lambda _{p^*}$ as $q\rightarrow p^*$ (if $p<N$), and that $\lambda _q,\Vert \nabla u_q\Vert _p\rightarrow +\infty$ as $q\rightarrow 0$ (if $|\Omega |<1$).

2 Preliminaries

In an attempt to make this paper as self-contained as possible, we will recall some well-known results. In the ordered Banach space $W^{1,p}_0(\Omega )$, the positive cone

$$\begin{aligned} W_+=\left\{ u\in W^{1,p}_0(\Omega ) \ : \ u\ge 0 \ \text{ a.e. } \text{ in } \Omega \right\} \end{aligned}$$

has empty interior. Instead, in $C^1_0(\overline{\Omega })$ the positive cone

$$\begin{aligned} C_+=\left\{ u\in C^1_0(\overline{\Omega }) \ : \ u(x)\ge 0 \quad \text{ for } \text{ all } \quad \ x\in \Omega \right\} \end{aligned}$$

has a nonempty interior, given by

$$\begin{aligned} \mathrm{int}(C_+)=\left\{ u\in C^1_0(\overline{\Omega }) \ : \ u(x)>0 \ \quad \text{ for } \text{ all } \quad x\in \Omega \hbox { and }\displaystyle \frac{\partial u}{\partial n}(x)<0\quad \hbox { for all} \quad x\in \partial \Omega \right\} \end{aligned}$$

(see Gasiński and Papageorgiou [10, Remark 6.2.10]). We consider the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _p u=f(u) &{}\text{ in } \Omega \\ u=0 &{}\text{ on } \partial \Omega \end{array},\right. \end{aligned}$$

(3)

where $f:{\mathbb R}\rightarrow ]0,+\infty [$ is a continuous function. We recall that $\underline{u}\in W^{1,p}(\Omega )$ is a sub-solution of problem (3) if $\underline{u}\le 0$ on $\partial \Omega$ (in the distributional sense) and

$$\begin{aligned} \int \limits _\Omega |\nabla \underline{u}|^{p-2}\nabla \underline{u}\cdot \nabla v \ \hbox {d}x\le \int \limits _\Omega f(\underline{u})v \ \hbox {d}x \ \quad \text{ for } \text{ all } \quad v\in W_+, \end{aligned}$$

Similarly, $\overline{u}$ is a super-solution of (3) if $\overline{u}\ge 0$ on $\partial \Omega$ and

$$\begin{aligned} \int \limits _\Omega |\nabla \overline{u}|^{p-2}\nabla \overline{u}\cdot \nabla v \ \hbox {d}x\ge \int \limits _\Omega f(\overline{u})v \ \hbox {d}x \ \quad \text{ for } \text{ all } \quad v\in W_+. \end{aligned}$$

We have the following result for sub- and super-solutions (a slightly rephrased version of Theorem 2.2 of Faria et al. [7]):

Theorem 2

If the mapping $t\mapsto t^{1-p}f(t)$ is decreasing in $]0,+\infty [$ and $\underline{u},\overline{u}\in W^{1,p}(\Omega )\cap C^{1,\beta }(\Omega )$ ($0<\beta \le 1$) are a positive sub-solution and a positive super-solution, respectively, of (3) s.t.

$$\begin{aligned} \underline{u},\overline{u}, \Delta _p\underline{u}, \Delta _p\overline{u}, \underline{u}/\overline{u}, \overline{u}/\underline{u}\in L^\infty (\Omega ), \end{aligned}$$

then $\underline{u}(x)\le \overline{u}(x)$ for all $x\in \Omega$.

We recall a consequence of the strong nonlinear maximum principle (see Vázquez [20, Theorem 5]):

Theorem 3

If $u\in C_+\setminus \{0\}, \Delta _p u\in L^2_\mathrm{loc}(\Omega )$ and $\Delta _p u\le 0$ a.e. in $\Omega$, then $u\in \mathrm{int}(C_+)$.

We denote by $\hat{u}_1\in \mathrm{int}(C_+)$ the unique positive solution of the constant right-hand side problem

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _p u=1 &{}\text{ in } \Omega \\ u=0 &{}\text{ on } \partial \Omega \end{array}.\right. \end{aligned}$$

(4)

3 Proof of the main result

We will split the proof of Theorem 1 in several steps. We begin by achieving some global properties of the mapping $g$:

Lemma 4

The mapping $g$ is Lipschitz continuous on compact subsets of $]0,p^*[$ and nonincreasing in $]0,p^*[$.

Proof

By $\partial B_1(0)$, we denote the unit sphere in $W^{1,p}_0(\Omega )$ centered at $0$. For all $u\in \partial B_1(0)$, the mapping $q\mapsto \Vert u\Vert _q^q$ is at least twice differentiable in $]0,p^*[$ with

$$\begin{aligned} \frac{\hbox {d}^2}{\hbox {d}q^2}\Vert u\Vert ^q_q=\int \limits _{\{u\ne 0\}}|u|^q(\ln |u|)^2 \ \hbox {d}x\ge 0 \quad \text{ for } \text{ all }\quad q\in ]0,p^{*}{[}, \end{aligned}$$

So, $q\mapsto \Vert u\Vert _q^q$ is convex for all $u\in \partial B_1(0)$. Now set

$$\begin{aligned} h(q)=\sup _{u\in \partial B_1(0)}\Vert u\Vert _q^q \quad \text{ for } \text{ all }\quad q\in ]0,p^{*}{[}. \end{aligned}$$

The mapping $h:]0,p^*[\rightarrow ]0,+\infty [$ is convex. By (2), it is easily seen that

$$\begin{aligned} g(q)=(h(q))^{-\frac{p}{q}}|\Omega |^\frac{p}{q}= e^{-\frac{p}{q}\log (|\Omega |^{-1}h(q))} \quad \text{ for } \text{ all } \quad q\in ]0,p^*[. \end{aligned}$$

(5)

Now fix a compact set $K\subset ]0,p^*[$. Being convex, $h$ is Lipschitz continuous in $K$. Since $\min _{q\in K}(|\Omega |^{-1}h(q))>0$, the mapping $q\mapsto \log (|\Omega |^{-1}h(q))$ is still Lipschitz continuous in $K$ and from (5), we easily deduce the first part of our claim.

Now, let $0<r<q<p^*$ be real numbers. For all $u\in \partial B_1(0)$, Hölder inequality yields

$$\begin{aligned} \Vert u\Vert _r^r\le \Vert u\Vert _q^r|\Omega |^\frac{q-r}{q}, \end{aligned}$$

so we have $h(r)\le h(q)^\frac{r}{q}|\Omega |^\frac{q-r}{q}$. Using (5), we easily obtain

$$\begin{aligned} g(r)\ge g(q). \end{aligned}$$

Thus, $g$ is nonincreasing in $]0,p^*[$. $\square$

In order to prove strict monotonicity of $g$, we need to consider separately the intervals $]0,1]$ and $]1,p^*[$. We first focus on the case $0<q\le 1$. We say $u\in W^{1,p}_0(\Omega )$ is a weak solution of problem (1) if

$$\begin{aligned} \int \limits _\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla v \ \hbox {d}x=\lambda \int \limits _\Omega |u|^{q-2}uv \ \hbox {d}x \end{aligned}$$

(6)

for all $v\in W^{1,p}_0(\Omega )$ with $\mathrm{supp}(v)\subset \Omega$. We have the following existence result:

Lemma 5

If $0<q\le 1$, then problem (1) with $\lambda =\lambda _q$ admits a positive weak solution $u_q\in W^{1,p}_0(\Omega )\cap C^1(\Omega )$ s.t.

• (i) $\Vert \nabla u_q\Vert _p=\lambda _q^\frac{1}{p}, \Vert u_q\Vert _q=1$;

• (ii) $c_1\hat{u}_1(x)\le u_q(x)\le c_2(1+\hat{u}_1(x))$ for all $x\in \Omega$ ($c_1,c_2>0$);

• (iii) if $0<r<q\le 1$, then $u_r\ne u_q$ on a dense subset of $\Omega$.

Proof

Let $(\varepsilon _n)$ be a decreasing sequence in $]0,1[$ s.t. $\varepsilon _n\rightarrow 0$ as $n\rightarrow \infty$. For all $n\in {\mathbb N}$, we consider the nonsingular problem

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _p u=(u+\varepsilon _n)^{q-1} &{}\quad \text{ in } \Omega \\ u\ge 0 &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ on } \partial \Omega \end{array}.\right. \end{aligned}$$

(7)

We set

$$\begin{aligned} \varphi _n(u)=\frac{\Vert \nabla u\Vert _p^p}{p}-\int \limits _\Omega \left[ \frac{1}{q}(u^++\varepsilon _n)^q-\varepsilon _n^{q-1}u^-\right] \hbox {d}x \ \text{ for } \text{ all } \quad u\in W^{1,p}_0(\Omega ) \end{aligned}$$

(as usual, we denote $t^\pm =\max \{\pm t,0\}$). It is easily seen that

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\left[ \frac{1}{q}(t^++\varepsilon _n)^q-\varepsilon _n^{q-1}t^-\right] =(t^++\varepsilon _n)^{q-1} \ \text{ for } \text{ all } \quad t\in {\mathbb R}. \end{aligned}$$

So, $\varphi _n\in C^1(W^{1,p}_0(\Omega ))$ and for all $u,v\in W^{1,p}_0(\Omega )$ we have

$$\begin{aligned} \langle \varphi '_n(u),v\rangle =\int \limits _\Omega |\nabla u|^{p-2}\nabla u\cdot \nabla v \ \hbox {d}x-\int \limits _\Omega (u^++\varepsilon _n)^{q-1}v \ \hbox {d}x. \end{aligned}$$

In particular, if $u\in W_+$ is a critical point of $\varphi _n$, then $u$ is a weak solution of (7). The functional $\varphi _n$ is coercive and sequentially weakly lower semicontinuous in $W^{1,p}_0(\Omega )$, so there exists $u_n\in W^{1,p}_0(\Omega )$ s.t.

$$\begin{aligned} \varphi _n(u_n)=\inf _{u\in W^{1,p}_0(\Omega )}\varphi _n(u). \end{aligned}$$

(8)

It is easily seen that $\varphi _n(u^+)\le \varphi _n(u)$ for all $u\in W^{1,p}_0(\Omega )$, hence we may assume $u_n\in W_+$. So, $u_n$ is a weak solution of (7). Moreover, nonlinear regularity theory (see Gasiński and Papageorgiou [10, Theorem 1.5.5], and Lieberman [15, Theorem 1]) implies $u_n\in C^{1,\beta }(\overline{\Omega })$ for some $\beta >0$. A standard argument shows $u_n\ne 0$. So, we can apply Theorem 3 and obtain $u_n\in \mathrm{int}(C_+)$.

Now we will prove some useful estimates for the function $u_n$. First, we observe that, for $\rho >0$ small enough, $\rho \hat{u}_1$ is a sub-solution of (7). Indeed, choose $\rho >0$ small s.t.

$$\begin{aligned} \rho ^{p-1}<(\rho \hat{u}_1(x)+1)^{q-1} \quad \text{ for } \text{ all } \quad x\in \Omega . \end{aligned}$$

Then, we have for all $v\in W_+$

$$\begin{aligned} \int \limits _\Omega |\nabla (\rho \hat{u}_1)|^{p-2}\nabla (\rho \hat{u}_1)\cdot \nabla v \ \hbox {d}x=\rho ^{p-1}\int \limits _\Omega v \ \hbox {d}x\le \int \limits _\Omega \left( \rho \hat{u}_1+\varepsilon _n\right) ^{q-1}v \ \hbox {d}x. \end{aligned}$$

Clearly we can regard $u_n$ as a super-solution of (7). By regularity theory, taking $\beta >0$ even smaller if necessary, we may assume $\rho \hat{u}_1,u_n\in C^{1,\beta }(\overline{\Omega })$. Moreover, since $u_n\in \mathrm{int}(C_+)$, there exists $r>0$ s.t. $u_n-r\rho \hat{u}_1\in \mathrm{int}(C_+)$, in particular

$$\begin{aligned} u_n(x)-r\rho \hat{u}_1(x)>0 \quad \text{ for } \text{ all } \quad x\in \Omega , \end{aligned}$$

which implies $\rho \hat{u}_1/u_n\in L^\infty (\Omega )$. Similarly we prove that $u_n/\rho \hat{u}_1\in L^\infty (\Omega )$. Thus, we apply Theorem 2 and we have

$$\begin{aligned} \rho \hat{u}_1(x)\le u_n(x) \ \text{ for } \text{ all } \quad x\in \Omega \quad (\hbox {with } \rho >0 \,\hbox {independent of} \,n). \end{aligned}$$

(9)

We set

$$\begin{aligned} \Omega _n=\left\{ x\in \Omega \ : \ u_n(x)>1\right\} . \end{aligned}$$

Clearly, $\Omega _n\subset \Omega$ is open with a $C^{1,\beta }$ boundary (due to the regularity of $u_n$). Without any loss of generality, we may assume that $\Omega _n$ is connected and consider the Dirichlet problem:

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _p u=(u+1+\varepsilon _n)^{q-1} &{}\quad \text{ in } \ \Omega _n \\ u\ge 0 &{}\quad \text{ in } \ \Omega _n \\ u=0 &{}\quad \text{ on } \ \partial \Omega _n \end{array}.\right. \end{aligned}$$

(10)

Therefore, $u_n-1$ is a positive sub-solution of (10). With an argument analogous to that employed above, we prove that $\hat{u}_1$ is a super-solution of (10). An application of Theorem 2 then yields

$$\begin{aligned} u_n(x)\le \hat{u}_1(x)+1 \ \quad \text{ for } \text{ all }\quad x\in \Omega . \end{aligned}$$

(11)

For all $n\in {\mathbb N}$, we have

$$\begin{aligned} \Vert \nabla u_n\Vert _p^p=\int \limits _\Omega (u_n+\varepsilon _n)^{q-1}u_n \ \hbox {d}x\le \Vert u_n\Vert _q^q\le \lambda _q^{-\frac{q}{p}}\Vert \nabla u_n\Vert _p^q \ \text{(see } \text{(2)), } \end{aligned}$$

hence the sequence $(u_n)$ is bounded in $W^{1,p}_0(\Omega )$. Passing to a subsequence, we may assume that there exists $\tilde{u}_q\in W^{1,p}_0(\Omega )$ s.t. $u_n\rightharpoonup \tilde{u}_q$ in $W^{1,p}_0(\Omega )$ and $u_n\rightarrow \tilde{u}_q$ in $L^q(\Omega )$. In particular, $u_n(x)\rightarrow \tilde{u}_q(x)$ for a.e. $x\in \Omega$, so $\tilde{u}_q\in W_+$.

We shall now prove that $u_n\rightarrow \tilde{u}_q$ in $W^{1,p}_0(\Omega )$. First we observe that, since $u_n\rightharpoonup \tilde{u}_q$ in $W^{1,p}_0(\Omega )$,

$$\begin{aligned} \liminf _{n}\frac{\Vert \nabla u_n\Vert _p^p}{p}\ge \frac{\Vert \nabla \tilde{u}_q\Vert _p^p}{p}. \end{aligned}$$

(12)

We set for all $u\in W^{1,p}_0(\Omega )$

$$\begin{aligned} \varphi _q(u)=\frac{\Vert \nabla u\Vert _p^p}{p}-\frac{\Vert u\Vert _q^q}{q}, \end{aligned}$$

so $\varphi _q:W^{1,p}_0(\Omega )\rightarrow {\mathbb R}$ is a continuous (though not differentiable) functional. It is easily seen that

$$\begin{aligned} \lim _n\varphi _n(u)=\varphi _q(u) \quad \text{ for } \text{ all }\quad u\in W_+. \end{aligned}$$

(13)

Also, from (8), we have for all $n\in {\mathbb N}$

$$\begin{aligned} \frac{\Vert \nabla u_n\Vert _p^p}{p}&= \varphi _n(u_n)+\frac{1}{q}\int \limits _\Omega (u_n+\varepsilon _n)^q \ \hbox {d}x \\&\le \varphi _n(\tilde{u}_q)+\frac{1}{q}\int \limits _\Omega (u_n+\varepsilon _n)^q \ \hbox {d}x. \end{aligned}$$

Hence, by (13) and since $u_n\rightarrow \tilde{u}_q$ in $L^q(\Omega )$,

$$\begin{aligned} \limsup _n\frac{\Vert \nabla u_n\Vert _p^p}{p}\le \varphi _q(\tilde{u}_q)+\frac{\Vert \tilde{u}_q\Vert _q^q}{q}=\frac{\Vert \nabla \tilde{u}_q\Vert _p^p}{p}, \end{aligned}$$

which, together with (12), gives $\Vert \nabla u_n\Vert _p\rightarrow \Vert \nabla \tilde{u}_q\Vert _p$. This in turn implies $u_n\rightarrow \tilde{u}_q$ in $W^{1,p}_0(\Omega )$.

Obviously, (9) and (11) imply

$$\begin{aligned} \rho \hat{u}_1\le \tilde{u}_q\le \hat{u}_1+1 \ \text{ a.e. } \text{ in } \Omega \text{. } \end{aligned}$$

(14)

We prove now that $\tilde{u}_q$ is a weak solution of problem (1) with $\lambda =1$. Let us fix $v\in W^{1,p}_0(\Omega )$ with $\mathrm{supp}(v)\subset \Omega$. We have for all $n\in {\mathbb N}$

$$\begin{aligned} \int \limits _\Omega |\nabla u_n|^{p-2}\nabla u_n\cdot \nabla v \ \hbox {d}x=\int \limits _\Omega (u_n+\varepsilon _n)^{q-1}v \ \hbox {d}x. \end{aligned}$$

By (9), (11) we can pass to the limit in the above equality as $n\rightarrow \infty$ and get

$$\begin{aligned} \int \limits _\Omega |\nabla \tilde{u}_q|^{p-2}\nabla \tilde{u}_q\cdot \nabla v \ \hbox {d}x=\int \limits _\Omega \tilde{u}_q^{q-1}v \ \hbox {d}x. \end{aligned}$$

(15)

Now we discuss regularity of $\tilde{u}_q$. For any smooth domain $\Omega '$ s.t. $\overline{\Omega }'\subset \Omega$, we have from (15)

$$\begin{aligned} \int \limits _{\Omega '}|\nabla \tilde{u}_q|^{p-2}\nabla \tilde{u}_q\cdot \nabla v \ \hbox {d}x=\int \limits _{\Omega '}\tilde{u}_q^{q-1}v \ \hbox {d}x \end{aligned}$$

for all $v\in W^{1,p}_0(\Omega )$ with $\mathrm{supp}(v)\subset \Omega '$. By interior regularity theory for degenerate elliptic equations (see Di Benedetto [5, Theorem 2]), we have $\tilde{u}_q\in C^1(\Omega ')$, which, as $\Omega '$ is arbitrary, implies $\tilde{u}_q\in C^1(\Omega )$.

It is easily seen that

$$\begin{aligned} \liminf _n\varphi _n(u_n)\ge \varphi _q(\tilde{u}_q). \end{aligned}$$

(16)

From (8), (13), and (16), we have for all $u\in W^{1,p}_0(\Omega )$

$$\begin{aligned} \varphi _q(u)&= \varphi _q(u^+)+\varphi _q(u^-) \\&= \lim _n\left( \varphi _n(u^+)+\varphi _n(u^-)\right) \\&= \lim _n\left[ \varphi _n(|u|)+\int \limits _{\{u=0\}}\frac{\varepsilon _n^q}{q}\hbox {d}x\right] \\&\ge \liminf _n\varphi _n(u_n) \\&\ge \varphi _q(\tilde{u}_q). \end{aligned}$$

So,

$$\begin{aligned} \varphi _q(\tilde{u}_q)=\inf _{u\in W^{1,p}_0(\Omega )}\varphi _q(u). \end{aligned}$$

(17)

Since $u_n$ solves (7), we have for all $n\in {\mathbb N}$

$$\begin{aligned} \Vert \nabla u_n\Vert _p^p=\int \limits _\Omega (u_n+\varepsilon _n)^{q-1}u_n \ \hbox {d}x. \end{aligned}$$

Passing again to the limit as $n\rightarrow \infty$, we have

$$\begin{aligned} \Vert \nabla \tilde{u}_q\Vert _p^p=\Vert \tilde{u}_q\Vert _q^q. \end{aligned}$$

(18)

We set

$$\begin{aligned} T=\left\{ w\in W ^{1,p}_0(\Omega ) \ : \ w\ne 0, \ \Vert \nabla w\Vert _p^p=\Vert w\Vert _q^q\right\} . \end{aligned}$$

From (17) and (18),

$$\begin{aligned} \left( \frac{1}{ p}-\frac{1}{q}\right) \Vert \tilde{u}_q\Vert _q^q\le \left( \frac{1}{p}-\frac{1}{q}\right) \Vert w\Vert _q^q \ \quad \text{ for } \text{ all } \quad w\in T, \end{aligned}$$

So, we have

$$\begin{aligned} \Vert \tilde{u}_q\Vert _q^q=\sup _{w\in T}\Vert w\Vert _q^q=\sup _{u\in W^{1,p}_0(\Omega ), \ u\ne 0}\left( \frac{\Vert u\Vert _q}{\Vert \nabla u\Vert _p}\right) ^\frac{pq}{p-q}=\lambda _q^\frac{q}{q-p}. \end{aligned}$$

Finally, set $u_q=\tilde{u}_q/\Vert \tilde{u}_q\Vert _q$. Clearly $u_q\in C^1(\Omega )$ is a weak solution of (1) with $\lambda =\lambda _q$. Moreover, $\Vert u_q\Vert _q=1$ and by (18) we also have $\Vert \nabla u_q\Vert _p=\lambda _q^\frac{1}{p}$, so $u_q$ satisfies $(i)$. By (14), we can find $c_1,c_2>0$ s.t. $(ii)$ holds. Finally, we prove $(iii)$: let $0<r<q\le 1$ be real numbers. We will prove that $u_r\ne u_q$ on a dense subset of $\Omega$, arguing by contradiction. Assume that $u_r=u_q$ in $A$, where $A\subset \Omega$ is a nonempty open set. For any $v\in W^{1,p}_0(\Omega )$ s.t. $\mathrm{supp}(v)\subset A$, we have by (6)

$$\begin{aligned} \lambda _q\int u_q^{q-1}v \ \hbox {d}x=\lambda _r\int \limits _A u_q^{r-1}v \ \hbox {d}x, \end{aligned}$$

which implies $u_q(x)=(\lambda _r/\lambda _q)^\frac{1}{q-r}$ for every $x\in A$, hence $\Delta _p u_q=0$ a.e. in $A$, a contradiction. So the proof is concluded. $\square$

Now we can prove strict monotonicity of $g$ in $]0,1]$:

Lemma 6

The mapping $g$ is decreasing in $]0,1]$.

Proof

We know from Lemma 4 that $g$ is nonincreasing. We argue by contradiction, assuming that there exist $0<r<q\le 1$ s.t. $g(r)=g(q)$, hence

$$\begin{aligned} \lambda _r=\lambda _q|\Omega |^{\frac{p}{q}-\frac{p}{r}}. \end{aligned}$$

By Lemma 5 (and rescaling), there exist $\check{u}_r,\check{u}_q\in \partial B_1(0), \check{u}_r\ne \check{u}_q$, s.t. $\Vert \check{u}_r\Vert _r=\lambda _r^{-\frac{1}{p}}$ and $\Vert \check{u}_q\Vert _q=\lambda _q^{-\frac{1}{p}}$. We apply Hölder inequality and the above equality to get

$$\begin{aligned} \lambda _q^{-\frac{r}{p}}|\Omega |^{1-\frac{r}{q}}=\Vert \check{u}_r\Vert _r^r\le \Vert \check{u}_r\Vert _q^r|\Omega |^{1-\frac{r}{q}}\le \lambda _q^{-\frac{r}{p}}|\Omega |^{1-\frac{r}{q}}, \end{aligned}$$

so $\Vert \check{u}_r\Vert _q=\lambda _q^{-\frac{1}{p}}$. By concavity of the functional $u\mapsto \Vert u\Vert _q^q$ (recall that $q\le 1$), we have

$$\begin{aligned} \left\| \frac{\check{u}_r+\check{u}_q}{2}\right\| _q^q\ge \lambda _q^{-\frac{q}{p}}. \end{aligned}$$

Setting

$$\begin{aligned} \check{w}=\frac{(\check{u}_r+\check{u}_q)/2}{\Vert \nabla (\check{u}_r+\check{u}_q)/2\Vert _p}, \end{aligned}$$

we obviously have $\check{w}\in \partial B_1(0)$ and

$$\begin{aligned} \Vert \check{w}\Vert _q>\left\| \frac{\check{u}_r+\check{u}_q}{2}\right\| _q\ge \lambda _q^{-\frac{1}{p}}, \end{aligned}$$

against (2).$\square$

We now turn to the case $1<q<p^*$: in this case problem (1) can be treated via purely variational methods.

Lemma 7

If $1<q<p^*$, then problem (1) with $\lambda =\lambda _q$ admits a solution $u_q\in \mathrm{int}(C_+)$ s.t.

• (i) $\Vert \nabla u_q\Vert _p=\lambda _q^\frac{1}{p}$ and $\Vert u_q\Vert _q=1$;

• (ii) if $1<r<q<p^*$, then $u_r\ne u_q$ on a dense subset of $\Omega$.

Proof

Set

$$\begin{aligned} \partial B_1^q(0)=\left\{ u\in W^{1,p}_0(\Omega ) \ : \ \Vert u\Vert _q^q=1\right\} . \end{aligned}$$

The set $\partial B_1^q(0)$ is sequentially weakly closed in $W^{1,p}_0(\Omega )$. We can rephrase (2) as follows:

$$\begin{aligned} \lambda _q=\inf _{u\in \partial B_1^q(0)}\Vert \nabla u\Vert _p^p. \end{aligned}$$

(19)

By standard variational arguments, there exists $u_q\in \partial B_1^q(0)$ s.t. $\Vert \nabla u_q\Vert _p^p=\lambda _q$. We may assume $u_q\in W_+$ (otherwise we pass to $|u_q|$). Lagrange multipliers theory on Finsler manifolds (see for instance Perera et al. [17], p. 65) implies that there exists $\mu \in {\mathbb R}\setminus \{0\}$ s.t.

$$\begin{aligned} \int \limits _\Omega |\nabla u_q|^{p-2}\nabla u_q\cdot \nabla v \ \hbox {d}x=\mu \int \limits _\Omega u_q^{q-1}v \ \hbox {d}x \quad \text{ for } \text{ all }\quad v\in W^{1,p}_0(\Omega ). \end{aligned}$$

Taking $v=u_q$, we have

$$\begin{aligned} \lambda _q=\Vert \nabla u_q\Vert _p^p=\mu \Vert u_q\Vert _q^q=\mu . \end{aligned}$$

So, $u_q$ is a weak solution of (1) with $\lambda =\lambda _q$ and satisfies $(i)$. Nonlinear regularity theory (see Gasiński and Papageorgiou [10, Theorem 1.5.5], and Lieberman [15, Theorem 1]) implies $u_q\in C_+$. Besides, since $u_q\in \partial B_1^q(0)$, we have $u_q\ne 0$. So, we can apply Theorem 3 and obtain $u_q\in \mathrm{int}(C_+)$.

As in Lemma 5, we can achieve $(ii)$. $\square$

We complete the proof of Theorem 1 by introducing the following result:

Lemma 8

The mapping $g$ is decreasing in $]1,p^*[$.

Proof

Arguing by contradiction, we assume that there exist $1<r<q<p^*$ s.t. $g(r)=g(q)$. Applying Lemma 7 (and rescaling), we deduce that there exists $\check{u}_r\in \partial B_1(0)\cap \mathrm{int}(C_+)$ s.t.

$$\begin{aligned} \Vert \check{u}_r\Vert _r^r=\lambda _r^{-\frac{r}{p}}=\sup _{u\in \partial B_1(0)}\Vert u\Vert _r^r. \end{aligned}$$

By Lagrange multipliers theory, there exists $\mu \in {\mathbb R}\setminus \{0\}$ s.t.

$$\begin{aligned} \int \limits _\Omega |\nabla \check{u}_r|^{p-2}\nabla \check{u}_r\cdot \nabla v \ \hbox {d}x=\mu \int \limits _\Omega \check{u}_r^{r-1}v \ \hbox {d}x \ \quad \text{ for } \text{ all } \quad v\in W^{1,p}_0(\Omega ). \end{aligned}$$

Since $g(r)=g(q)$, arguing as in Lemma 6, we get

$$\begin{aligned} \Vert \check{u}_r\Vert _q^q=\lambda _q^{-\frac{q}{p}}=\sup _{u\in \partial B_1(0)}\Vert u\Vert _q^q. \end{aligned}$$

Hence, there exists $\nu \in {\mathbb R}\setminus \{0\}$ s.t.

$$\begin{aligned} \int \limits _\Omega |\nabla \check{u}_r|^{p-2}\nabla \check{u}_r\cdot \nabla v \ \hbox {d}x=\nu \int \limits _\Omega \check{u}_r^{q-1}v \ \hbox {d}x \ \quad \text{ for } \text{ all } \quad v\in W^{1,p}_0(\Omega ). \end{aligned}$$

This implies

$$\begin{aligned} \check{u}_r(x)=\left( \frac{\mu }{\nu }\right) ^\frac{1}{q-r} \quad \text{ for } \text{ all }\quad x\in \Omega , \end{aligned}$$

hence $\Delta _p\check{u}_r=0$ a.e. in $\Omega$, a contradiction. $\square$

Now, Theorem 1 is proved simply patching together Lemmas 4 (which implies, in particular, that $g$ is continuous in $]0,p^*[$), 6 and 8.

4 Further results

In this final section, we will examine the asymptotic behavior of the mapping $g$ as $q$ approaches either $0$ or $p^*$. First, we prove that (if $p<N$) $g$ admits a continuous extension to $]0,p^*]$. According to the definition of $g$, we set

$$\begin{aligned} g(p^*)=\lambda _{p^*}|\Omega |^\frac{p}{p^*}. \end{aligned}$$

Theorem 9

If $p<N$, then $g:]0,p^*]\rightarrow {\mathbb R}$ is absolutely continuous on compact subsets of $]0,p^*]$ and decreasing in $]0,p^*]$

Proof

First, we prove that

$$\begin{aligned} \lim _{q\rightarrow p^*}g(q)=\lambda _{p^*}|\Omega |^\frac{p}{p^*}. \end{aligned}$$

(20)

Let us fix $p<q<p^*$. By Lemma 7, there exists $u_q\in \mathrm{int}(C_+)$ s.t. $\Vert \nabla u_q\Vert _p^p=\lambda _q$ and $\Vert u_q\Vert _q^q=1$. We define $\varphi _q:W^{1,p}_0(\Omega )\rightarrow {\mathbb R}$ as in Lemma 5. This time we have $\varphi _q\in C^1(W^{1,p}_0(\Omega ))$, and all critical points of $\varphi _q$ are weak solutions of (1) with $\lambda =1$. It is easily seen that $\varphi _q(0)=0$ and $0$ is a strict local minimizer of $\varphi _q$. Besides, for all $\delta >0$ we have

$$\begin{aligned} \varphi _q(\delta u_q)=\frac{\lambda _q\delta ^p}{p}-\frac{\delta ^q}{q}, \end{aligned}$$

so $\varphi _q(\delta u_q)\rightarrow -\infty$ as $\delta \rightarrow +\infty$. So, we take $\delta >1$ s.t. $\varphi _q(\delta u_q)<0$ and set

$$\begin{aligned} \Gamma&= \left\{ \gamma \in C([0,\delta ],W^{1,p}_0(\Omega )) \ : \ \gamma (0)=0, \ \gamma (\delta )=\delta u_q\right\} , \\ c&= \inf _{\gamma \in \Gamma }\max _{t\in [0,\delta ]}\varphi _q(\gamma (t))>0. \end{aligned}$$

By the mountain pass theorem of Ambrosetti and Rabinowitz [1], there exists a critical point $\overline{u}_q\in W^{1,p}_0(\Omega )$ of $\varphi _q$ s.t. $\varphi _q(\overline{u}_q)=c$. In particular, we have

$$\begin{aligned} \Vert \nabla \overline{u}_q\Vert _p^p=\Vert \overline{u}_q\Vert _q^q>0. \end{aligned}$$

(21)

We determine the value in (21). By (2), we have

$$\begin{aligned} \lambda _q^\frac{1}{p}\le \frac{\Vert \nabla \overline{u}_q\Vert _p}{\Vert \overline{u}_q\Vert _q}=\Vert \nabla \overline{u}_q\Vert _p^\frac{q-p}{q}. \end{aligned}$$

Moreover, straightforward computation and (21) lead to

$$\begin{aligned} \left( \frac{1}{p}-\frac{1}{q}\right) \Vert \nabla \overline{u}_q\Vert _p^p&= \varphi _q(\overline{u}_q) \\&\le \max _{t\in [0,\delta ]}\varphi _q(t u_q) \\&= \left( \frac{1}{p}-\frac{1}{q}\right) \lambda _q^\frac{q}{q-p}. \end{aligned}$$

So, we perfect (21) getting

$$\begin{aligned} \Vert \nabla \overline{u}_q\Vert _p^p=\Vert \overline{u}_q\Vert _q^q=\lambda _q^\frac{q}{q-p}. \end{aligned}$$

(22)

By Garcia Azorero and Peral Alonso [9, Lemma 5], we have

$$\begin{aligned} \lim _{q\rightarrow p^*}\varphi _q(\overline{u}_q)=\frac{\lambda _{p^*}^\frac{N}{p}}{N}, \end{aligned}$$

which, together with (22), yields

$$\begin{aligned} \lim _{q\rightarrow p^*}\lambda _q=\lim _{q\rightarrow p^*}\left[ \frac{pq}{q-p}\varphi _q(\overline{u}_q)\right] ^\frac{q-p}{q}=\lambda _{p^*}. \end{aligned}$$

Thus, we get (20).

So, $g$ extends to a continuous, decreasing mapping on $]0,p^*]$, which we still denote $g$. We already know from Lemma 4 that, for all $0<a<b<p^*, g$ is Lipschitz (in particular, absolutely continuous) in $[a,b]$. Moreover, there exists $g'$ almost everywhere in $]0,p^*[$ and for all $0<a<b<p^*$ we have

$$\begin{aligned} \int \limits _a^b g'(q)dq=g(b)-g(a). \end{aligned}$$

(23)

In fact, we can pass to the limit in (23) as $b\rightarrow p^*$. Indeed, the left-hand side tends to $\int \nolimits _a^{p^*}g'(q)dq$ by basic results in measure theory, while the right-hand side tends to $g(p^*)-g(a)$ by (20). So we get

$$\begin{aligned} \int \limits _a^{p^*}g'(q)dq=g(p^*)-g(a). \end{aligned}$$

By classical results in real analysis (see, for instance, Royden and Fitzpatrick [18, Section 6.5,Corollary 12]), $g$ turns out to be absolutely continuous in $[a,p^*]$. This concludes the proof. $\square$

Remark 10

In [6], Ercole, assuming $p<N$, proves that the map $q\mapsto \lambda _q$ is of bounded variation in $[1,p^*]$ (this follows from the monotonicity of $g$), Lipschitz continuous in any closed interval of the type $[1, p^*-\varepsilon ]$ for $\varepsilon >0$ and left-side continuous at $q=p^*$. Combining these properties, the author obtains that $\lambda _q$ is absolutely continuous on $[1,p^*]$. The techniques adopted in [6] rely on some formula which describes the dependence of the Rayleigh quotient with respect to the parameter $q$. Our result extends those of [6] in a twofold sense: $p$ is allowed to be also greater or equal than $N$, and when $p<N, \lambda _q$ is absolutely continuous in a bigger interval than $[1,p^*]$.

As said in the Introduction, we can apply our results to the study of asymptotic behavior of $\lambda _q$ and $u_q$, in the spirit of Lee [14]:

Corollary 11

If $|\Omega |<1$, then

$$\begin{aligned} \lim _{q\rightarrow 0}\lambda _q=\lim _{q\rightarrow 0}\Vert \nabla u_q\Vert _p=+\infty . \end{aligned}$$

Proof

We clearly have $|\Omega |^\frac{p}{q}\rightarrow 0$ as $q\rightarrow 0$. By Theorem 1

$$\begin{aligned} \lim _{q\rightarrow 0}g(q)=\sup _{0<q<p^*}g(q)>0, \end{aligned}$$

from which the thesis immediately follows. $\square$

We end our study by presenting an open problem: if $p\ge N$, what happens when $q\rightarrow +\infty$? Perhaps the properties of the mapping $g$ can be used, as in Corollary 11, to answer such question.