Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent

Abstract

In this paper, we study the following problem:

$$\{\begin{array}{c}-{\mathrm{\Delta }}_{p}u=\lambda {|u|}^{p-2}u+{|u|}^{p^{\ast }-2}u\text{in }\mathrm{\Omega },\hfill \\ u=0\text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $1<p<N$, $-{\mathrm{\Delta }}_{p}u=div({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ is the p-Laplacian, $p^{\ast }=pN/(N-p)$ is the critical Sobolev exponent and $\lambda >0$ is a parameter. By establishing a new deformation lemma, we show that if $N>p^2+p$, then for each $\lambda >0$, this problem has infinitely many sign-changing solutions, which extends the results obtained in (Cao et al. in J. Funct. Anal. 262: 2861-2902, 2012; Schechter and Zou in Arch. Ration. Mech. Anal. 197: 337-356, 2010).

1 Introduction

In this paper, we consider the following problem:

$$\{\begin{array}{c}-{\mathrm{\Delta }}_{p}u=\lambda {|u|}^{p-2}u+{|u|}^{p^{\ast }-2}u\text{in }\mathrm{\Omega },\hfill \\ u=0\text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

(1.1)

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) is a smooth bounded domain, $1<p<N$, $-{\mathrm{\Delta }}_{p}u=div({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ is the p-Laplacian, $p^{\ast }=pN/(N-p)$ is the critical Sobolev exponent and $\lambda >0$ is a parameter.

The first existence result of Problem (1.1) with $p=2$ was obtained by Brezis and Nirenberg in the celebrated paper [1]. In that paper, the authors proved that Problem (1.1) had a positive solution for $N\ge 4$ and $\lambda \in (0,{\lambda }_1^{\ast })$ or $N=3$ and $\lambda \in ({\lambda }_1^{\ast }/4,{\lambda }_1^{\ast })$, where ${\lambda }_1^{\ast }$ is the first eigenvalue of $(-\mathrm{\Delta },H_0^1(\mathrm{\Omega }))$. After that, many existence results have appeared for (1.1); one can see, for example, [2–7] and the references therein for case of $p=2$ and [8–11] and the references therein for case of $1<p<N$. In particular, in the case of $p=2$, the authors in [2] proved that the number of solutions of Problem (1.1) is bounded below by the number of eigenvalues of $(-\mathrm{\Delta },H_0^1(\mathrm{\Omega }))$ lying in the open interval $(\lambda ,\lambda +S{|\mathrm{\Omega }|}^{-2/N})$, where S is the best Sobolev constant and $|\mathrm{\Omega }|$ is the Lebesgue measure of Ω. In [5], the existence of infinitely many sign-changing solutions of (1.1) with $p=2$ has been obtained when $N\ge 4$, $\lambda >0$ and Ω is a ball, while it has been shown in [6] that (1.1) with $p=2$ has infinitely many sign-changing radial solutions when $N\ge 7$, $\lambda >0$ and Ω also is a ball. We remark that the methods used in [5, 6] are strongly dependent on the symmetry of the ball Ω. For a general bounded smooth domain Ω, by the method of invariant sets of the descending flow, the authors in [7] have shown that (1.1) with $p=2$ has infinitely many sign-changing solutions when $N\ge 7$ and $\lambda >0$, which extends the main result in [4].

The main purpose of this paper is to try to obtain the existence of infinitely many sign-changing solutions of Problem (1.1) for general $p\in (1,N)$. In a very recent work [9], the authors have proved that (1.1) has infinitely many solutions for $\lambda >0$ and $N>p^2+p$. However, by using the Picone identity (cf. [12, 13]), we see that every nonzero solution of Problem (1.1) is sign-changing for $\lambda \ge {\lambda }_1$, where ${\lambda }_1$ is the first eigenvalue of $(-{\mathrm{\Delta }}_p,W_0^{1,p}(\mathrm{\Omega }))$ (see Lemma 2.1 for more details). Hence, to achieve our purpose, we mainly consider the situation of $\lambda \in (0,{\lambda }_1)$.

Our main result in this paper is the following.

Theorem 1.1 Assume that $N>p^2+p$ and $\lambda >0$. Then Problem (1.1) has infinitely many sign-changing solutions.

Since $p^{\ast }$ is the critical Sobolev exponent, in order to overcome the lack of compactness of the embedding of $W_0^{1,p}(\mathrm{\Omega })$ in the Lebesgue space $L^{p^{\ast }}(\mathrm{\Omega })$, we follow the ideas of [4, 7, 9] to consider the following auxiliary problems:

where $p_n<p^{\ast }$ and $p_n$ is increasing to $p^{\ast }$. It has been shown by [[14], Theorem 1.2] that for every n, Problem (${\mathcal{P}}_n$) has infinitely many sign-changing solutions ${\{u_{n,k}\}}_{k\in \mathbb{N}}$. Hence, to prove Theorem 1.1, we will show that for every $k\in \mathbb{N}$, $\{u_{n,k}\}$ converges to some sign-changing solution $u_k$ of (1.1) as $n\to \mathrm{\infty }$, and that $\{u_k\}$ are different. The convergence of $\{u_{n,k}\}$ can be done with the help of [[9], Theorem 1.2], which we show in Lemma 2.3. To distinguish $\{u_k\}$, we shall establish a new deformation lemma on special sets in $W_0^{1,p}(\mathrm{\Omega })$; see Lemma 2.5 for details.

Throughout this paper, we will always respectively denote $\parallel u\parallel$ and ${\parallel u\parallel }_r$ by the usual norm in spaces $W_0^{1,p}(\mathrm{\Omega })$ and $L^r(\mathrm{\Omega })$ ($r\ge 1$). Let C be indiscriminately used to denote various positive constants.

2 Proof of Theorem 1.1

We first consider the case of $\lambda \ge {\lambda }_1$. Recall that ${\lambda }_1$, the first eigenvalue of $-{\mathrm{\Delta }}_p$ in $W_0^{1,p}(\mathrm{\Omega })$, given by ${\lambda }_1:=inf\{{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p}dx,{\int }_{\mathrm{\Omega }}{|u|}^{p}dx=1\}$, is simple and there exists a positive eigenfunction $e_1\in W_0^{1,p}(\mathrm{\Omega })$ corresponding to ${\lambda }_1$ such that ${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_1|}^{p-2}\mathrm{\nabla }{e}_1\mathrm{\nabla }\eta dx={\lambda }_1{\int }_{\mathrm{\Omega }}{e}_1^{p-1}\eta dx$ for every $\eta \in W_0^{1,p}(\mathrm{\Omega })$ (cf. [15]). Moreover, by [[16], Proposition 2.1], we know that $e_1\in L^{\mathrm{\infty }}(\mathrm{\Omega })\cap C_{\mathrm{loc}}^{1,\alpha }(\mathrm{\Omega })$. On the other hand, we have the following proposition which is the so-called Picone identity.

Proposition 2.1 [[13], Lemma A.6]

Let $u,v\in W_{\mathrm{loc}}^{1,p}(\mathrm{\Omega })\cap C(\mathrm{\Omega })$ be such that $u\ge 0$, $v>0$ and $\frac{u}{v}\in W_{\mathrm{loc}}^{1,p}(\mathrm{\Omega })$. Then

$$\begin{array}{r}{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\left(\frac{u^p}{v^{p-1}}\right){|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }vdx\\ ={\int }_{\mathrm{\Omega }}(p{\left(\frac{u}{v}\right)}^{p-1}{|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v\mathrm{\nabla }u-(p-1){\left(\frac{u}{v}\right)}^p{|\mathrm{\nabla }v|}^p)dx.\end{array}$$

Moreover,

$${\int }_{\mathrm{\Omega }}(p{\left(\frac{u}{v}\right)}^{p-1}{|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v\mathrm{\nabla }u-(p-1){\left(\frac{u}{v}\right)}^p{|\mathrm{\nabla }v|}^p)dx\le {\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p}dx,$$

and the equality holds if and only if $u=cv$ for some constant $c>0$.

Lemma 2.1 Assume that $u\in W_0^{1,p}(\mathrm{\Omega })$ is a nonzero solution of (1.1) for $\lambda \ge {\lambda }_1$. Then u is sign-changing.

Proof By a contradiction, we may assume $u\ge 0$. By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have $u\in C^{1,\alpha }(\mathrm{\Omega })$ for some $\alpha \in (0,1)$. Thus, it follows from the strong maximum principle (cf. [18]) that $u>0$. Now, for every $\epsilon >0$, by applying the above Picone identity (i.e., Proposition 2.1) to $u+\epsilon$ and $e_1$, we see

$${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_1|}^{p}dx\ge {\int }_{\mathrm{\Omega }}\mathrm{\nabla }\left(\frac{e_1^p}{{(u+\epsilon )}^{p-1}}\right){|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }udx.$$

Noting that u is a solution of (1.1), we have

$$\begin{array}{r}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_1|}^{p}dx\ge {\int }_{\mathrm{\Omega }}(\lambda \frac{u^{p-1}}{{(u+\epsilon )}^{p-1}}+\frac{u^{p^{\ast }-1}}{{(u+\epsilon )}^{p-1}})e_1^{p}dx.\end{array}$$

It follows from the Fatou lemma that

$$\begin{array}{rl}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_1|}^{p}dx& \ge \underset{\epsilon \to 0}{lim\hspace{0.17em}inf}{\int }_{\mathrm{\Omega }}(\lambda \frac{u^{p-1}}{{(u+\epsilon )}^{p-1}}+\frac{u^{p^{\ast }-1}}{{(u+\epsilon )}^{p-1}})e_1^{p}dx\\ \ge {\int }_{\mathrm{\Omega }}(\lambda +u^{p^{\ast }-p})e_1^{p}dx,\end{array}$$

which is impossible since ${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_1|}^{p}dx={\lambda }_1{\int }_{\mathrm{\Omega }}{e}_1^p$, $u>0$, $e_1>0$ and $\lambda \ge {\lambda }_1$. Therefore, we have proved Lemma 2.1. □

Next, we consider the case of $\lambda <{\lambda }_1$.

It is clear that the corresponding functional of (${\mathcal{P}}_n$) $I_n:W_0^{1,p}(\mathrm{\Omega })\to \mathbb{R}$, given by

$$I_n(u)=\frac{1}{p}({\parallel u\parallel }^p-\lambda {\parallel u\parallel }_p^p)-\frac{1}{p_n}{\parallel u\parallel }_{p_n}^{p_n},$$

is $C^1$ Fréchet differentiable. Let $X_m=span\{{\phi }_1,\dots ,{\phi }_m\}$, where $\{{\phi }_i\}$ is a linearly independent sequence of $W_0^{1,p}(\mathrm{\Omega })$. It is easy to show that there exists $R_m>0$ such that $I_n(u)\le -1$ for $u\in X_m\mathrm{\setminus }{B}_m$, where $B_m:=\{u\in X_m:\parallel u\parallel \le R_m\}$ (cf. [[14], Lemma 3.9]). We denote

$$\begin{array}{c}P(-P):=\{u\in W_0^{1,p}(\mathrm{\Omega }):u\ge 0(u\le 0)\text{ a.e.}\},\hfill \\ D_{\mu }^{\pm }:=\{u\in W_0^{1,p}(\mathrm{\Omega }):dist(u,\pm P)\le \mu \},D_{\mu }:=D_{\mu }^{+}\cup D_{\mu }^{-},\hfill \\ G_m:=\{h\in C(B_m,W_0^{1,p}(\mathrm{\Omega })):h\text{ is odd, }h(x)=x\text{ for }x\in {\partial }_{X_m}{B}_m\}.\hfill \end{array}$$

Recall that the genus of a symmetric set A of $W_0^{1,p}(\mathrm{\Omega })$ is defined by

$$gen(A):=inf\{k\ge 0:\mathrm{\exists }f\in C(A,{\mathbb{R}}^k\mathrm{\setminus }\{0\}),f\text{ is odd}\}.$$

Here, we say that A is symmetric if $x\in A$ implies $-x\in A$.

By [[14], Theorem 1.2], we know that, for every $n\in \mathbb{N}$, $I_n(u)$ has infinitely many critical points, denoted by ${\{u_{n,k}\}}_{k\in \mathbb{N}}$, in $X\mathrm{\setminus }{D}_{\mu }$ for μ small enough. Moreover,

$$I_n(u_{n,k})=d_{n,k}:=\underset{Z\in {\mathrm{\Gamma }}_k}{inf}\underset{u\in Z}{sup}{I}_n(u),$$

(2.1)

where ${\mathrm{\Gamma }}_k:=\{h(B_m\mathrm{\setminus }B)\mathrm{\setminus }{D}_{\mu }:h\in G_m\text{ for }m\ge n,B=-B\subset B_m\text{ open,}gen(B)\le m-n\}$.

Lemma 2.2 For every $k\in \mathbb{N}$, there exists $d_k^{\ast }>0$ such that $\parallel u_{n,k}\parallel \le d_k^{\ast }$ for all $n\in \mathbb{N}$.

Proof Consider the following auxiliary functional:

$$I_{\ast }(u):=\frac{1}{p}({\parallel u\parallel }^p-{\parallel u\parallel }_p^p)-\frac{1}{p^{\ast }}{\parallel u\parallel }_{\sigma }^{\sigma },$$

where $\sigma =(p+p^{\ast })/2$. Since $p_n\to p^{\ast }$, we may assume $p_n>\sigma$ for all $n\in \mathbb{N}$. Then $\frac{1}{p^{\ast }}{\parallel u\parallel }_{\sigma }^{\sigma }\le \frac{meas(\mathrm{\Omega })}{p^{\ast }}+\frac{1}{p_n}{\parallel u\parallel }_{p_n}^{p_n}$ for all $n\in \mathbb{N}$. This means

$$I_{\ast }(u)=I_n(u)+(\frac{1}{p_n}{\parallel u\parallel }_{p_n}^{p_n}-\frac{1}{p^{\ast }}{\parallel u\parallel }_{\sigma }^{\sigma })\ge I_n(u)-\frac{meas(\mathrm{\Omega })}{p^{\ast }}.$$

(2.2)

Note that $I_{\ast }(u)$ is the corresponding functional of the following equation:

$$\begin{array}{r}\{\begin{array}{c}-{\mathrm{\Delta }}_{p}u=\lambda {|u|}^{p-2}u+\frac{\sigma }{p^{\ast }}{|u|}^{\sigma -2}u\text{in }\mathrm{\Omega },\hfill \\ u=0\text{on }\partial \mathrm{\Omega },\hfill \end{array}\end{array}$$

and the nonlinearity satisfies the assumptions of [[14], Theorem 1.2]. Thus, this equation has a sequence of solutions $\{v_k\}\subset W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }(D_{\mu }^{+}\cup D_{\mu }^{-})$ such that

$$I_{\ast }(v_k)={\overline{d}}_k:=\underset{Z\in {\mathrm{\Gamma }}_k}{inf}\underset{u\in Z}{sup}{I}_{\ast }(u)$$

for μ small enough. For every $k\in \mathbb{N}$, the definitions of ${\overline{d}}_k$ and $d_{k,n}$, together with (2.2), imply ${\overline{d}}_k+\frac{meas(\mathrm{\Omega })}{p^{\ast }}\ge d_{k,n}$ for all $n\in \mathbb{N}$. On the other hand, since for every n, ${\{u_{n,k}\}}_{k\in \mathbb{N}}$ is a sequence of solutions for (${\mathcal{P}}_n$) whose energies satisfy (2.1), it follows that $d_{n,k}\ge (\frac{1}{p}-\frac{1}{p_1})(1-\frac{\lambda }{{\lambda }_1}){\parallel u_{n,k}\parallel }^p$. We complete the proof by choosing $d_k^{\ast }={(\frac{({\overline{d}}_k{p}^{\ast }+meas(\mathrm{\Omega }))p_{1}p{\lambda }_1}{p^{\ast }(p_1-p)({\lambda }_1-\lambda )})}^{1/p}$. □

By Lemma 2.2 and [[9], Theorem 1.2], we know that for each $k\in \mathbb{N}$, there exists $u_k\in W_0^{1,p}(\mathrm{\Omega })$ such that $u_{n,k}\to u_k$ as $n\to \mathrm{\infty }$ in $W_0^{1,p}(\mathrm{\Omega })$. The next lemma will give more information about $u_k$.

Lemma 2.3 $u_k$ is a sign-changing solution of Problem (1.1) for every $k\in \mathbb{N}$.

Proof We first prove that $u_k$ is a solution of Problem (1.1) for every $k\in \mathbb{N}$. Since $u_{n,k}\to u_k$ as $n\to \mathrm{\infty }$ in $W_0^{1,p}(\mathrm{\Omega })$,

$${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_{n,k}|}^{p-2}\mathrm{\nabla }{u}_{n,k}\mathrm{\nabla }\phi dx\to {\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_k|}^{p-2}\mathrm{\nabla }{u}_k\mathrm{\nabla }\phi dx$$

and

$${\int }_{\mathrm{\Omega }}{|u_{n,k}|}^{p-2}{u}_{n,k}\phi dx\to {\int }_{\mathrm{\Omega }}{|u_k|}^{p-2}{u}_k\phi dx$$

as $n\to \mathrm{\infty }$ for every $\phi \in W_0^{1,p}(\mathrm{\Omega })$. If we can prove

$${\int }_{\mathrm{\Omega }}{|u_{n,k}|}^{p_n-2}{u}_{n,k}\phi dx\to {\int }_{\mathrm{\Omega }}{|u_k|}^{2^{\ast }-2}{u}_k\phi dx$$

(2.3)

as $n\to \mathrm{\infty }$ for every $\phi \in W_0^{1,p}(\mathrm{\Omega })$, then $u_k$ is a solution of (1.1) for $u_{n,k}$ is a solution of (${\mathcal{P}}_n$). Indeed, $u_{n,k}\to u_k$ a.e. in Ω as $n\to \mathrm{\infty }$ since $u_{n,k}\to u_k$ in $W_0^{1,p}(\mathrm{\Omega })$. By the Egoroff theorem, for every $\delta >0$, there exists ${\mathrm{\Omega }}_{\delta }$ such that $u_{n,k}\to u_k$ uniformly in $\mathrm{\Omega }\mathrm{\setminus }{\mathrm{\Omega }}_{\delta }$ and $|{\mathrm{\Omega }}_{\delta }|<\delta$, where $|{\mathrm{\Omega }}_{\delta }|$ is the Lebesgue measure of ${\mathrm{\Omega }}_{\delta }$. This, together with the Lebesgue dominated convergence theorem, implies

$$\underset{n\to \mathrm{\infty }}{lim}{\int }_{\mathrm{\Omega }\mathrm{\setminus }{\mathrm{\Omega }}_{\delta }}{|u_{n,k}|}^{p_n-2}{u}_{n,k}\phi dx={\int }_{\mathrm{\Omega }\mathrm{\setminus }{\mathrm{\Omega }}_{\delta }}{|u_k|}^{p^{\ast }-2}{u}_k\phi dx\text{for every }\phi \in W_0^{1,p}(\mathrm{\Omega }).$$

(2.4)

On the other hand, for every $\phi \in W_0^{1,p}(\mathrm{\Omega })$, we have

$$\begin{array}{r}{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_{n,k}|}^{p_n-2}{u}_{n,k}-{|u_k|}^{p^{\ast }-2}{u}_k||\phi |dx\\ \le {\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_{n,k}|}^{p_n-2}{u}_{n,k}-{|u_{n,k}|}^{p-1}-{|u_{n,k}|}^{p^{\ast }-1}||\phi |dx\\ +{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_k|}^{p^{\ast }-2}{u}_k||\phi |dx\\ +{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_{n,k}|}^{p-1}-{|u_{n,k}|}^{p^{\ast }-1}||\phi |dx\\ \le 2{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_{n,k}|}^{p-1}+{|u_{n,k}|}^{p^{\ast }-1}||\phi |dx+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_k|}^{p^{\ast }-2}{u}_k||\phi |dx\\ +{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_{n,k}|}^{p-1}-{|u_{n,k}|}^{p^{\ast }-1}||\phi |dx\\ \le 2{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}||\phi |dx+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_k|}^{p^{\ast }-2}{u}_k||\phi |dx\\ +3{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_{n,k}|}^{p-1}-{|u_{n,k}|}^{p^{\ast }-1}||\phi |dx.\end{array}$$

For every $\epsilon >0$, by the above inequality and the absolute continuity of the integral, we can take δ small enough such that

$$2{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}||\phi |dx+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_k|}^{p^{\ast }-2}{u}_k||\phi |dx<\epsilon /3.$$

For this δ, since $u_{n,k}\to u_k$ in $W_0^{1,p}(\mathrm{\Omega })$,

$$3{\int }_{{\mathrm{\Omega }}_{\delta }}|{|u_k|}^{p-1}+{|u_k|}^{p^{\ast }-1}-{|u_{n,k}|}^{p-1}-{|u_{n,k}|}^{p^{\ast }-1}||\phi |dx<\epsilon /3$$

for n large enough. By (2.4), for this δ, we have

$${\int }_{\mathrm{\Omega }\mathrm{\setminus }{\mathrm{\Omega }}_{\delta }}|{|u_{n,k}|}^{p_n-2}{u}_{n,k}\phi -{|u_k|}^{p^{\ast }-2}{u}_k\phi |dx<\epsilon /3$$

for n large enough. So (2.3) holds. Moreover, by a similar proof, we can show $d_k:={lim}_{n\to \mathrm{\infty }}{d}_{n,k}=I_n(u_{n,k})=I(u_k)$.

Next, we will show $u_k$ is sign-changing for all $k\in \mathbb{N}$. Since for each $n\in \mathbb{N}$, $u_{n,k}$ is a sign-changing solution of (${\mathcal{P}}_n$), multiplying (${\mathcal{P}}_n$) by $u_{n,k}^{\pm }$, we obtain ${\parallel u_{n,k}^{\pm }\parallel }^p=\lambda {\parallel u_{n,k}^{\pm }\parallel }_p^p+{\parallel u_{n,k}^{\pm }\parallel }_{p_n}^{p_n}$, where $u^{\pm }=max\{\pm u,0\}$. Note that $\lambda <{\lambda }_1$, by the Sobolev imbedding theorem, we have $0<(1-\frac{\lambda }{{\lambda }_1})C\le {\parallel u_{n,k}^{\pm }\parallel }_{p^{\ast }}^{p_n-p}$. It follows that $u_{n,k}\to u_k$ in $L^{p^{\ast }}(\mathrm{\Omega })$ as $n\to \mathrm{\infty }$ for $u_{n,k}\to u_k$ in $W_0^{1,p}(\mathrm{\Omega })$ as $n\to \mathrm{\infty }$. This gives $0<(1-\frac{\lambda }{{\lambda }_1})C\le {\parallel u_k^{\pm }\parallel }_{p^{\ast }}^{p^{\ast }-p}$, i.e., $u_k^{\pm }\ne 0$ for all $k\in \mathbb{N}$. □

Let $\epsilon >0$ and $c\in \mathbb{R}$, we denote

$$\begin{array}{c}K:=\{u\in W_0^{1,p}(\mathrm{\Omega }):I^{\prime }(u)=0\},K_c:=\{u\in W_0^{1,p}(\mathrm{\Omega }):I(u)=c,I^{\prime }(u)=0\},\hfill \\ K_{\mu }^{\ast }:=K\mathrm{\setminus }(int\left(D_{\mu }^{+}\right)\cup int\left(D_{\mu }^{-}\right)),K_{c,\mu }^{\ast }:=K_c\mathrm{\setminus }(int\left(D_{\mu }^{+}\right)\cup int\left(D_{\mu }^{-}\right)),\hfill \\ {\mathcal{N}}_{c,\mu ,\epsilon }:=\{u\in W_0^{1,p}(\mathrm{\Omega }):dist(u,K_c^{\ast })<\epsilon \}.\hfill \end{array}$$

Thanks to Lemma 2.3, $u_k\in K_{{\mu }_k}^{\ast }$ for some ${\mu }_k>0$. We claim that $\{u_k\}\subset K_{\mu }^{\ast }$ for some $\mu >0$. Indeed, if not, then $dist(u_k,P)\to 0$ as $k\to \mathrm{\infty }$ without loss of generality. On the one hand, since $u_k$ is a solution of (1.1), ${〈I^{\prime }(u_k),u_k-S_{\lambda }(u_k)〉}_{W_0^{1,p}(\mathrm{\Omega }),W_0^{-1,p}(\mathrm{\Omega })}=0$, where $S_{\lambda }(u_k):{(-{\mathrm{\Delta }}_p)}^{-1}(\lambda {|u_k|}^{p-2}{u}_k+{|u_k|}^{p^{\ast }-2}{u}_k)$. On the other hand, by [[17], Lemma 3.7], we have

$${〈I^{\prime }(u_k),u_k-S_{\lambda }(u_k)〉}_{W_0^{1,p}(\mathrm{\Omega }),W_0^{-1,p}(\mathrm{\Omega })}\ge C{\parallel u_k-S_{\lambda }(u_k)\parallel }^2{(\parallel u_k\parallel +\parallel S_{\lambda }(u_k)\parallel )}^{p-2}$$

for $1<p<2$ and

$${〈I^{\prime }(u_k),u_k-S_{\lambda }(u_k)〉}_{W_0^{1,p}(\mathrm{\Omega }),W_0^{-1,p}(\mathrm{\Omega })}\ge C{\parallel u_k-S_{\lambda }(u_k)\parallel }^p$$

for $p\ge 2$. Note that by a similar proof of [[14], Lemma 3.3], we can see that $S_{\lambda }(D_{\mu }^{\pm })\subset int(D_{\mu }^{\pm })$ for μ small enough. Thus, $\parallel u_k-S_{\lambda }(u_k)\parallel >0$ for k large enough. This implies

$${〈I^{\prime }(u_k),u_k-S_{\lambda }(u_k)〉}_{W_0^{1,p}(\mathrm{\Omega }),W_0^{-1,p}(\mathrm{\Omega })}\ge C_k>0$$

for k large enough, which contradicts ${〈I^{\prime }(u_k),u_k-S_{\lambda }(u_k)〉}_{W_0^{1,p}(\mathrm{\Omega }),W_0^{-1,p}(\mathrm{\Omega })}=0$. For the sake of convenience, we denote $K_{\mu }^{\ast }$, $K_{c,\mu }^{\ast }$, ${\mathcal{N}}_{c,\mu ,\epsilon }$ by $K^{\ast }$, $K_c^{\ast }$, ${\mathcal{N}}_{c,\epsilon }$. Note that for every $c\in \mathbb{R}$, $K_c$ is compact in $W_0^{1,p}(\mathrm{\Omega })$ (cf. [[9], Theorem 1.2]). It follows from [[19], Proposition 7.5] that there exists $\epsilon >0$ such that

$$gen({\mathcal{N}}_{c,2\epsilon })=gen\left(K_c^{\ast }\right)<+\mathrm{\infty }.$$

(2.5)

Let $J_n^c:=\{u\in W_0^{1,p}(\mathrm{\Omega }):I_n(u)\le c\}$ and ${\mathcal{Q}}_n^c:=D_{\mu }\cup J_n^c$. Let $J^c:=\{u\in W_0^{1,p}(\mathrm{\Omega }):I(u)\le c\}$. For $\delta >0$ small enough, we define ${\mathcal{A}}_{n,\epsilon }^{c,\delta }:=({\mathcal{Q}}_n^{c+\delta }\mathrm{\setminus }{\mathcal{Q}}_n^{c-\delta })\mathrm{\setminus }{\mathcal{N}}_{c,\epsilon }$, then we have the following.

Lemma 2.4 Assume that there exists $\delta >0$ such that $K^{\ast }\cap J^{c+\delta }\mathrm{\setminus }int(J^{c-\delta })=K_c^{\ast }$ for n large. Then there exists $\gamma >0$ such that $\parallel I_n^{\prime }(u)\parallel \ge \gamma$ for $u\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ and large n.

Proof Assume a contradiction. Then, for every $n\in \mathbb{N}$, there exists $\{v_{n,k}\}\subset {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ such that ${lim}_{k\to \mathrm{\infty }}{I}_n^{\prime }(v_{n,k})=0$. It is clear that $I_n$ satisfies the (PS) condition for every $n\in \mathbb{N}$. Hence there exists $v_n\in W_0^{1,p}(\mathrm{\Omega })$ such that, up to a subsequence, $v_{n,k}\to v_n$ in $W_0^{1,p}(\mathrm{\Omega })$ as $k\to \mathrm{\infty }$ with $I_n^{\prime }(v_n)=0$ and $I_n(v_n)\in [c-\delta ,c+\delta ]$. This implies

$$c+\delta \ge I_n(v_n)=(\frac{1}{p}-\frac{1}{p_n})(1-\frac{\lambda }{{\lambda }_1}){\parallel v_n\parallel }^p\ge (\frac{1}{p}-\frac{1}{p_1})(1-\frac{\lambda }{{\lambda }_1}){\parallel v_n\parallel }^p.$$

Thus, by [[9], Theorem 1.2], up to a subsequence, we see that there exists $v_0\in W_0^{1,p}(\mathrm{\Omega })$ such that $v_n\to v_0$ in $W_0^{1,p}(\mathrm{\Omega })$ as $n\to \mathrm{\infty }$. Moreover, by using the arguments in the proof of Lemma 2.3, we have $I^{\prime }(v_0)=0$ and $I(v_0)\in [c-\delta ,c+\delta ]$. On the other hand, for large n, $v_n\notin (int(D_{\mu }^{+})\cup int(D_{\mu }^{-}))\cup {\mathcal{N}}_{c,\epsilon }$ since $v_{n,k}\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$. It follows that $v_0\notin (int(D_{\mu }^{+})\cup int(D_{\mu }^{-}))\cup {\mathcal{N}}_{c,\epsilon }$. This contradicts the fact that $K^{\ast }\cap J_n^{c+\delta }\mathrm{\setminus }int(J_n^{c-\delta })=K_c^{\ast }$. □

Lemma 2.5 Assume that there exists $\gamma >0$ such that $\parallel I_n^{\prime }(u)\parallel \ge \gamma$ for every $u\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ and large n. Then there exist $\delta >0$ and an odd continuous map ${\eta }_n$ such that ${\eta }_n:{\mathcal{A}}_{n,2\epsilon }^{c,\delta }\cup {\mathcal{Q}}_n^{c-\delta }\to {\mathcal{Q}}_n^{c-\delta }$ and $\eta {|}_{{\mathcal{Q}}_n^{c-\delta }}=\mathit{Id}$ for large n.

Proof We first assume $1<p<2$. It is clear that there exists $L>0$ such that

$$\parallel u\parallel +\parallel S_{n,\lambda }(u)\parallel \le L\text{for all }u\in {\mathcal{N}}_{c,2\epsilon },$$

(2.6)

where

$$〈S_{n,\lambda }(u),\phi 〉:={\int }_{\mathrm{\Omega }}(\lambda {|u|}^{p-2}u+{|u|}^{p_n-2}u)\phi dx\text{for }u\in W_0^{1,p}(\mathrm{\Omega })\text{ and }\phi \in W_0^{-1,p}(\mathrm{\Omega }).$$

Let $T_{n,\lambda }:W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }K\to W_0^{1,p}(\mathrm{\Omega })$ be the local Lipschitz continuous operator obtained in [[14], Lemma 2.1] and let ${\varphi }_u(t)$ be the solution of the following O.D.E.

$$\{\begin{array}{c}\frac{d\varphi }{dt}=-\varphi +T_{n,\lambda }(\varphi ),\hfill \\ \varphi =u\in W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }K.\hfill \end{array}$$

Denote $\tau (u)$ to be the maximal interval of existence of ${\varphi }_u(t)$.

Claim 1: ${\varphi }_u(t)$ cannot enter ${\mathcal{N}}_{c,\epsilon }$ before it enters ${\mathcal{Q}}_n^{c-\delta }$ for small δ, large n and $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$.

Indeed, if the claim fails, then for every $\delta >0$, ${\varphi }_u(t)$ will enter ${\mathcal{N}}_{c,\epsilon }$ before it enters ${\mathcal{Q}}_n^{c-\delta }$. Since $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }\subset W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }{\mathcal{N}}_{c,2\epsilon }$, there exist $0\le t_1<t_2<\tau (u)$ such that ${\varphi }_u(t)\in {\mathcal{N}}_{c,2\epsilon }\mathrm{\setminus }{\mathcal{N}}_{c,\epsilon }$ for $t\in (t_1,t_2]$ and

$$dist({\varphi }_u(t_1),K_c^{\ast })=2\epsilon ,dist({\varphi }_u(t_2),K_c^{\ast })=\epsilon .$$

By [[14], Lemma 2.1], $C{\parallel u-S_{n,\lambda }(u)\parallel }^2{(\parallel u\parallel +\parallel S_{n,\lambda }(u)\parallel )}^{p-2}\le 〈I_n(u),u-T_{n,\lambda }(u)〉$. On the other hand, by the choice of $t_1$ and $t_2$, we know that ${\varphi }_u(t)\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ for $t\in (t_1,t_2]$. Thanks to [[17], Lemma 3.8], $\parallel u-S_{n,\lambda }(u)\parallel \ge {(\frac{\gamma }{C})}^{1/(p-1)}$ for large n. This, together with (2.6) and [[14], Lemma 2.1], implies

$$\begin{array}{rl}\epsilon & \le \parallel {\varphi }_u(t_2)-{\varphi }_u(t_1)\parallel \le {\int }_{t_1}^{t_2}\parallel {\varphi }_u(t)-T_{n,\lambda }({\varphi }_u(t))\parallel dt\\ \le C{\int }_{t_1}^{t_2}\parallel {\varphi }_u(t)-S_{n,\lambda }({\varphi }_u(t))\parallel dt\\ \le C{\int }_{t_1}^{t_2}{\parallel {\varphi }_u(t)-S_{n,\lambda }({\varphi }_u(t))\parallel }^2{(\parallel {\varphi }_u(t)\parallel +\parallel S_{n,\lambda }({\varphi }_u(t))\parallel )}^{p-2}dt\\ \le C{\int }_{t_1}^{t_2}〈I_n({\varphi }_u(t)),{\varphi }_u(t)-T_{n,\lambda }({\varphi }_u(t))〉dt\\ =C(I_n(t_1)-I_n(t_2))\le 4C\delta .\end{array}$$

A contradiction with $\delta <4C/\epsilon$.

Claim 2: There exists ${\tau }_1(t)<\tau (u)$ such that ${\varphi }_u({\tau }_1(u))\in {\mathcal{Q}}_n^{c-\delta }$ for large n and $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$.

If the claim is not true, then ${\varphi }_u(t)\in {\mathcal{Q}}_n^{c+\delta }\mathrm{\setminus }{\mathcal{Q}}_n^{c-\delta }$ for all $t\in (0,\tau (u))$. We first consider the case of $\tau (u)<+\mathrm{\infty }$. In fact, by Claim 1, ${\varphi }_u(t)\notin {\mathcal{N}}_{c,\epsilon }$, i.e., ${\varphi }_u(t)\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ for all $t\in (0,\tau (u))$. Since $\parallel I_n^{\prime }(u)\parallel \ge \gamma >0$ for $u\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ and large n, we must have

$$\parallel {\varphi }_u(t)\parallel \to \mathrm{\infty }\text{as }t\to \tau (u).$$

(2.7)

On the other hand, by [[14], Lemma 2.1] and [[17], Lemma 5.2], we have

$$\begin{array}{rl}\parallel {\varphi }_u(t)-{\varphi }_u(0)\parallel & \le {\int }_0^t\parallel {\varphi }_u(s)-T_{\lambda ,n}({\varphi }_u(s))\parallel ds\\ \le C{\int }_0^t\parallel {\varphi }_u(s)-S_{\lambda ,n}({\varphi }_u(s))\parallel ds\\ \le C{\int }_0^t{(1+\parallel {\varphi }_u(s)-S_{\lambda ,n}({\varphi }_u(s))\parallel )}^{p}ds\\ \le C{\int }_0^t{(1+\parallel {\varphi }_u(s)-S_{\lambda ,n}({\varphi }_u(s))\parallel )}^2{(\parallel {\varphi }_u(s)\parallel +\parallel S_{\lambda ,n}({\varphi }_u(s))\parallel )}^{p-2}ds\\ \le C{\int }_0^t{\parallel {\varphi }_u(s)-S_{\lambda ,n}({\varphi }_u(s))\parallel }^2{(\parallel {\varphi }_u(s)\parallel +\parallel S_{\lambda ,n}({\varphi }_u(s))\parallel )}^{p-2}ds\\ \le C(I_n({\varphi }_u(0))-I_n({\varphi }_u(t)))\le C.\end{array}$$

This means $\parallel {\varphi }_u(t)\parallel \le \parallel u\parallel +C$ for all $t\in (0,\tau (u))$, which contradicts with (2.7). It follows that there must exist ${\tau }_1(u)<\tau (u)$ such that ${\varphi }_u({\tau }_1(u))\in {\mathcal{Q}}_n^{c-\delta }$ for $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$, large n and $\tau (u)<+\mathrm{\infty }$. Next, we consider the case of $\tau (u)=+\mathrm{\infty }$. Since $\parallel u-S_{n,\lambda }(u)\parallel \ge {(\frac{\gamma }{C})}^{1/(p-1)}$ for all $u\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ and large n, it follows from [[14], Lemma 2.1] and [[17], Lemma 5.2] that

$$\begin{array}{rl}\frac{d{I}_n({\varphi }_u(t))}{dt}& =〈I_n({\varphi }_u(t)),-{\varphi }_u(t)+T_{n,\lambda }({\varphi }_u(t))〉\\ \le -C{\parallel {\varphi }_u(t)-S_{n,\lambda }({\varphi }_u(t))\parallel }^2{(\parallel {\varphi }_u(t)\parallel +\parallel S_{n,\lambda }({\varphi }_u(t))\parallel )}^{p-2}\\ \le -C{\parallel {\varphi }_u(t)-S_{n,\lambda }({\varphi }_u(t))\parallel }^2{(1+\parallel {\varphi }_u(t)-S_{n,\lambda }({\varphi }_u(t))\parallel )}^{p-2}\\ \le -C<0.\end{array}$$

Thus, there also exists ${\tau }_1(u)\in (0,+\mathrm{\infty })$ such that ${\varphi }_u({\tau }_1(u))\in {\mathcal{Q}}_n^{c-\delta }$ for $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$ and $\tau (u)=+\mathrm{\infty }$. Moreover, we must have ${\varphi }_u(t)\in {\mathcal{Q}}_n^{c-\delta }$ for $t\in ({\tau }_1(u),\tau (u))$ since $\frac{d{I}_n({\varphi }_u(t))}{dt}\le 0$ for all $u\in W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }K$.

Let

$${\eta }_n(u)=\{\begin{array}{cc}{\varphi }_u({\tau }_1(u)),\hfill & u\in {\mathcal{A}}_{n,2\epsilon }^c,\hfill \\ u,\hfill & u\in {\mathcal{Q}}_n^{c-\delta }.\hfill \end{array}$$

Then, by the continuity of ${\varphi }_u(t)$, ${\eta }_n(u)$ is continuous. Note that ${\varphi }_u(t)$ is odd and ${\tau }_1(u)$ is even, we see that ${\eta }_n(u)$ is odd and it is the desired map. The situation of $p\ge 2$ can be proved in a similar way. Therefore, we complete the proof of this lemma. □

Proof of Theorem 1.1 We first consider the case $\lambda \ge {\lambda }_1$. Thanks to Lemma 2.1 and [[9], Theorem 1.1], (1.1) has infinitely many sign-changing solutions. Next, we consider the case of $\lambda \in (0,{\lambda }_1)$. Since for every $n\in \mathbb{N}$, $0\le d_{n,k}\le d_{n,k+1}$ for all $k\in \mathbb{N}$, $d_k\le d_{k+1}$ for all $k\in \mathbb{N}$. It follows that two cases may occur:

Case 1: There are $1<k_1<k_2<\cdots$ such that $d_{k_1}<d_{k_2}<\cdots$ .

In this case, Problem (1.1) has infinitely many sign-changing solutions.

Case 2: There exists $k_0>0$ such that $d_{\ast }=d_k$ for all $k\ge k_0$.

In this case, if $(K^{\ast }\cap J^{d_{\ast }+\delta }\mathrm{\setminus }{J}^{d_{\ast }-\delta })\mathrm{\setminus }{K}_{d_{\ast }}^{\ast }\ne \mathrm{\varnothing }$ for every $\delta >0$ small enough, then Problem (1.1) also has infinitely many sign-changing solutions. Otherwise, there exists ${\delta }_0>0$ such that $(K^{\ast }\cap J^{d_{\ast }+\delta }\mathrm{\setminus }{J}^{d_{\ast }-\delta })=K_{d_{\ast }}^{\ast }$ for $\delta <{\delta }_0$. Thanks to Lemmas 2.4 and 2.5, there exists ${\eta }_n$ such that ${\eta }_n({\mathcal{A}}_{n,2\epsilon }^{d_{\ast }}\cup {\mathcal{Q}}_n^{d_{\ast }-\delta })\subset {\mathcal{Q}}_n^{d_{\ast }-\delta }$ for small δ and large n. Fix $l\in \mathbb{N}$ and $k\ge k_0$, the definitions of $d_k$ and $d_{k+l}$ give that there exists a large n such that $d_{n,k}>d_{\ast }-\delta$ and $d_{n,k+l}<d_{\ast }+\delta$ for small $\delta \in (0,1)$. By the definition of $d_{n,k+l}$, there exists $Z\in {\mathrm{\Gamma }}_{k+l}$ such that ${sup}_Z{I}_n(u)<d_{\ast }+\delta$, where $Z=h(B_m\mathrm{\setminus }B)\mathrm{\setminus }{D}_{\mu }$, $h\in G_m$ and $gen(B)\le m-k-l$. It follows that $h(B_m\mathrm{\setminus }B)\mathrm{\setminus }{\mathcal{N}}_{d_{\ast },2\epsilon }\subset {\mathcal{A}}_{n,2\epsilon }^{c,\delta }\cup {\mathcal{Q}}_n^{c-\delta }$. Thus, ${\eta }_n(h(B_m\mathrm{\setminus }B)\mathrm{\setminus }{\mathcal{N}}_{d_{\ast },2\epsilon })\subset {\mathcal{Q}}_n^{d_{\ast }-\delta }$. By the choice of δ and $B_m$, we have ${\eta }_n\circ h\in G_m$. If $gen(B\cup h^{-1}({\mathcal{N}}_{d_{\ast },2\epsilon }))\le m-k$, then we have

$$d_{\ast }-\delta <d_{n,k}\le \underset{{\eta }_n\circ h(B_m\mathrm{\setminus }(B\cup h^{-1}({\mathcal{N}}_{d_{\ast },2\epsilon })))}{sup}{I}_n(u)\le d_{\ast }-\delta .$$

A contradiction. By the properties of gen, we have

$$m-k+1\le gen(B\cup h^{-1}({\mathcal{N}}_{d_{\ast },2\epsilon }))\le gen(B)+gen({\mathcal{N}}_{d_{\ast },2\epsilon })\le m-k-l+gen({\mathcal{N}}_{d_{\ast },2\epsilon }).$$

This implies $gen({\mathcal{N}}_{d_{\ast },2\epsilon })\ge l+1$. Since $l\in \mathbb{N}$ is arbitrary, we have $gen({\mathcal{N}}_{d_{\ast },2\epsilon })=+\mathrm{\infty }$, which contradicts with (2.5). □