## Boundary Value Problems

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

Boundary Value Problems20132013:149

https://doi.org/10.1186/1687-2770-2013-149

Accepted: 28 May 2013

Published: 19 June 2013

## Abstract

In this paper, we study the following problem:

where $\mathrm{\Omega }\subset {\mathbb{R}}^{N}$ is a smooth bounded domain, $1, $-{\mathrm{\Delta }}_{p}u=div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)$ is the p-Laplacian, ${p}^{\ast }=pN/\left(N-p\right)$ 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:
(1.1)

where $\mathrm{\Omega }\subset {\mathbb{R}}^{N}$ ($N\ge 3$) is a smooth bounded domain, $1, $-{\mathrm{\Delta }}_{p}u=div\left({|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\right)$ is the p-Laplacian, ${p}^{\ast }=pN/\left(N-p\right)$ 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 $\lambda \in \left(0,{\lambda }_{1}^{\ast }\right)$ or $N=3$ and $\lambda \in \left({\lambda }_{1}^{\ast }/4,{\lambda }_{1}^{\ast }\right)$, where ${\lambda }_{1}^{\ast }$ is the first eigenvalue of $\left(-\mathrm{\Delta },{H}_{0}^{1}\left(\mathrm{\Omega }\right)\right)$. After that, many existence results have appeared for (1.1); one can see, for example, [27] and the references therein for case of $p=2$ and [811] and the references therein for case of $1. 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 $\left(-\mathrm{\Delta },{H}_{0}^{1}\left(\mathrm{\Omega }\right)\right)$ lying in the open interval $\left(\lambda ,\lambda +S{|\mathrm{\Omega }|}^{-2/N}\right)$, 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 $p\in \left(1,N\right)$. 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 $\left(-{\mathrm{\Delta }}_{p},{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\right)$ (see Lemma 2.1 for more details). Hence, to achieve our purpose, we mainly consider the situation of $\lambda \in \left(0,{\lambda }_{1}\right)$.

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 ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ in the Lebesgue space ${L}^{{p}^{\ast }}\left(\mathrm{\Omega }\right)$, 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 ${\left\{{u}_{n,k}\right\}}_{k\in \mathbb{N}}$. Hence, to prove Theorem 1.1, we will show that for every $k\in \mathbb{N}$, $\left\{{u}_{n,k}\right\}$ converges to some sign-changing solution ${u}_{k}$ of (1.1) as $n\to \mathrm{\infty }$, and that $\left\{{u}_{k}\right\}$ are different. The convergence of $\left\{{u}_{n,k}\right\}$ can be done with the help of [[9], Theorem 1.2], which we show in Lemma 2.3. To distinguish $\left\{{u}_{k}\right\}$, we shall establish a new deformation lemma on special sets in ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$; 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 ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ and ${L}^{r}\left(\mathrm{\Omega }\right)$ ($r\ge 1$). 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 ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, given by ${\lambda }_{1}:=inf\left\{{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx,{\int }_{\mathrm{\Omega }}{|u|}^{p}\phantom{\rule{0.2em}{0ex}}dx=1\right\}$, is simple and there exists a positive eigenfunction ${e}_{1}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ corresponding to ${\lambda }_{1}$ such that ${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_{1}|}^{p-2}\mathrm{\nabla }{e}_{1}\mathrm{\nabla }\eta \phantom{\rule{0.2em}{0ex}}dx={\lambda }_{1}{\int }_{\mathrm{\Omega }}{e}_{1}^{p-1}\eta \phantom{\rule{0.2em}{0ex}}dx$ for every $\eta \in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ (cf. [15]). Moreover, by [[16], Proposition 2.1], we know that ${e}_{1}\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)\cap {C}_{\mathrm{loc}}^{1,\alpha }\left(\mathrm{\Omega }\right)$. On the other hand, we have the following proposition which is the so-called Picone identity.

Proposition 2.1 [[13], Lemma A.6]

Let $u,v\in {W}_{\mathrm{loc}}^{1,p}\left(\mathrm{\Omega }\right)\cap C\left(\mathrm{\Omega }\right)$ be such that $u\ge 0$, $v>0$ and $\frac{u}{v}\in {W}_{\mathrm{loc}}^{1,p}\left(\mathrm{\Omega }\right)$. Then
$\begin{array}{r}{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\left(\frac{{u}^{p}}{{v}^{p-1}}\right){|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}={\int }_{\mathrm{\Omega }}\left(p{\left(\frac{u}{v}\right)}^{p-1}{|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v\mathrm{\nabla }u-\left(p-1\right){\left(\frac{u}{v}\right)}^{p}{|\mathrm{\nabla }v|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
Moreover,
${\int }_{\mathrm{\Omega }}\left(p{\left(\frac{u}{v}\right)}^{p-1}{|\mathrm{\nabla }v|}^{p-2}\mathrm{\nabla }v\mathrm{\nabla }u-\left(p-1\right){\left(\frac{u}{v}\right)}^{p}{|\mathrm{\nabla }v|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx,$

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

Lemma 2.1 Assume that $u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ 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 $u\in {C}^{1,\alpha }\left(\mathrm{\Omega }\right)$ for some $\alpha \in \left(0,1\right)$. 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
${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dx\ge {\int }_{\mathrm{\Omega }}\mathrm{\nabla }\left(\frac{{e}_{1}^{p}}{{\left(u+\epsilon \right)}^{p-1}}\right){|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\phantom{\rule{0.2em}{0ex}}dx.$
Noting that u is a solution of (1.1), we have
$\begin{array}{r}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dx\ge {\int }_{\mathrm{\Omega }}\left(\lambda \frac{{u}^{p-1}}{{\left(u+\epsilon \right)}^{p-1}}+\frac{{u}^{{p}^{\ast }-1}}{{\left(u+\epsilon \right)}^{p-1}}\right){e}_{1}^{p}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
It follows from the Fatou lemma that
$\begin{array}{rl}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dx& \ge \underset{\epsilon \to 0}{lim inf}{\int }_{\mathrm{\Omega }}\left(\lambda \frac{{u}^{p-1}}{{\left(u+\epsilon \right)}^{p-1}}+\frac{{u}^{{p}^{\ast }-1}}{{\left(u+\epsilon \right)}^{p-1}}\right){e}_{1}^{p}\phantom{\rule{0.2em}{0ex}}dx\\ \ge {\int }_{\mathrm{\Omega }}\left(\lambda +{u}^{{p}^{\ast }-p}\right){e}_{1}^{p}\phantom{\rule{0.2em}{0ex}}dx,\end{array}$

which is impossible since ${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{e}_{1}|}^{p}\phantom{\rule{0.2em}{0ex}}dx={\lambda }_{1}{\int }_{\mathrm{\Omega }}{e}_{1}^{p}$, $u>0$, ${e}_{1}>0$ 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 (${\mathcal{P}}_{n}$) ${I}_{n}:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\to \mathbb{R}$, given by
${I}_{n}\left(u\right)=\frac{1}{p}\left({\parallel u\parallel }^{p}-\lambda {\parallel u\parallel }_{p}^{p}\right)-\frac{1}{{p}_{n}}{\parallel u\parallel }_{{p}_{n}}^{{p}_{n}},$
is ${C}^{1}$ Fréchet differentiable. Let ${X}_{m}=span\left\{{\phi }_{1},\dots ,{\phi }_{m}\right\}$, where $\left\{{\phi }_{i}\right\}$ is a linearly independent sequence of ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. It is easy to show that there exists ${R}_{m}>0$ such that ${I}_{n}\left(u\right)\le -1$ for $u\in {X}_{m}\mathrm{\setminus }{B}_{m}$, where ${B}_{m}:=\left\{u\in {X}_{m}:\parallel u\parallel \le {R}_{m}\right\}$ (cf. [[14], Lemma 3.9]). We denote
Recall that the genus of a symmetric set A of ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ is defined by

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 $n\in \mathbb{N}$, ${I}_{n}\left(u\right)$ has infinitely many critical points, denoted by ${\left\{{u}_{n,k}\right\}}_{k\in \mathbb{N}}$, in $X\mathrm{\setminus }{D}_{\mu }$ for μ small enough. Moreover,
${I}_{n}\left({u}_{n,k}\right)={d}_{n,k}:=\underset{Z\in {\mathrm{\Gamma }}_{k}}{inf}\underset{u\in Z}{sup}{I}_{n}\left(u\right),$
(2.1)

where .

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:
${I}_{\ast }\left(u\right):=\frac{1}{p}\left({\parallel u\parallel }^{p}-{\parallel u\parallel }_{p}^{p}\right)-\frac{1}{{p}^{\ast }}{\parallel u\parallel }_{\sigma }^{\sigma },$
where $\sigma =\left(p+{p}^{\ast }\right)/2$. Since ${p}_{n}\to {p}^{\ast }$, we may assume ${p}_{n}>\sigma$ for all $n\in \mathbb{N}$. Then $\frac{1}{{p}^{\ast }}{\parallel u\parallel }_{\sigma }^{\sigma }\le \frac{meas\left(\mathrm{\Omega }\right)}{{p}^{\ast }}+\frac{1}{{p}_{n}}{\parallel u\parallel }_{{p}_{n}}^{{p}_{n}}$ for all $n\in \mathbb{N}$. This means
${I}_{\ast }\left(u\right)={I}_{n}\left(u\right)+\left(\frac{1}{{p}_{n}}{\parallel u\parallel }_{{p}_{n}}^{{p}_{n}}-\frac{1}{{p}^{\ast }}{\parallel u\parallel }_{\sigma }^{\sigma }\right)\ge {I}_{n}\left(u\right)-\frac{meas\left(\mathrm{\Omega }\right)}{{p}^{\ast }}.$
(2.2)
Note that ${I}_{\ast }\left(u\right)$ is the corresponding functional of the following equation:
and the nonlinearity satisfies the assumptions of [[14], Theorem 1.2]. Thus, this equation has a sequence of solutions $\left\{{v}_{k}\right\}\subset {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\mathrm{\setminus }\left({D}_{\mu }^{+}\cup {D}_{\mu }^{-}\right)$ such that
${I}_{\ast }\left({v}_{k}\right)={\overline{d}}_{k}:=\underset{Z\in {\mathrm{\Gamma }}_{k}}{inf}\underset{u\in Z}{sup}{I}_{\ast }\left(u\right)$

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 ${\overline{d}}_{k}+\frac{meas\left(\mathrm{\Omega }\right)}{{p}^{\ast }}\ge {d}_{k,n}$ for all $n\in \mathbb{N}$. On the other hand, since for every n, ${\left\{{u}_{n,k}\right\}}_{k\in \mathbb{N}}$ is a sequence of solutions for (${\mathcal{P}}_{n}$) whose energies satisfy (2.1), it follows that ${d}_{n,k}\ge \left(\frac{1}{p}-\frac{1}{{p}_{1}}\right)\left(1-\frac{\lambda }{{\lambda }_{1}}\right){\parallel {u}_{n,k}\parallel }^{p}$. We complete the proof by choosing ${d}_{k}^{\ast }={\left(\frac{\left({\overline{d}}_{k}{p}^{\ast }+meas\left(\mathrm{\Omega }\right)\right){p}_{1}p{\lambda }_{1}}{{p}^{\ast }\left({p}_{1}-p\right)\left({\lambda }_{1}-\lambda \right)}\right)}^{1/p}$. □

By Lemma 2.2 and [[9], Theorem 1.2], we know that for each $k\in \mathbb{N}$, there exists ${u}_{k}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that ${u}_{n,k}\to {u}_{k}$ as $n\to \mathrm{\infty }$ in ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. 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 ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$,
${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_{n,k}|}^{p-2}\mathrm{\nabla }{u}_{n,k}\mathrm{\nabla }\phi \phantom{\rule{0.2em}{0ex}}dx\to {\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{u}_{k}|}^{p-2}\mathrm{\nabla }{u}_{k}\mathrm{\nabla }\phi \phantom{\rule{0.2em}{0ex}}dx$
and
${\int }_{\mathrm{\Omega }}{|{u}_{n,k}|}^{p-2}{u}_{n,k}\phi \phantom{\rule{0.2em}{0ex}}dx\to {\int }_{\mathrm{\Omega }}{|{u}_{k}|}^{p-2}{u}_{k}\phi \phantom{\rule{0.2em}{0ex}}dx$
as $n\to \mathrm{\infty }$ for every $\phi \in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. If we can prove
${\int }_{\mathrm{\Omega }}{|{u}_{n,k}|}^{{p}_{n}-2}{u}_{n,k}\phi \phantom{\rule{0.2em}{0ex}}dx\to {\int }_{\mathrm{\Omega }}{|{u}_{k}|}^{{2}^{\ast }-2}{u}_{k}\phi \phantom{\rule{0.2em}{0ex}}dx$
(2.3)
as $n\to \mathrm{\infty }$ for every $\phi \in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, 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 ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$. 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
(2.4)
On the other hand, for every $\phi \in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, we have
$\begin{array}{r}{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{n,k}|}^{{p}_{n}-2}{u}_{n,k}-{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le {\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{n,k}|}^{{p}_{n}-2}{u}_{n,k}-{|{u}_{n,k}|}^{p-1}-{|{u}_{n,k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{n,k}|}^{p-1}-{|{u}_{n,k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le 2{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{n,k}|}^{p-1}+{|{u}_{n,k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{n,k}|}^{p-1}-{|{u}_{n,k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{1em}{0ex}}\le 2{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}||\phi |\phantom{\rule{0.2em}{0ex}}dx\\ \phantom{\rule{2em}{0ex}}+3{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{n,k}|}^{p-1}-{|{u}_{n,k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
For every $\epsilon >0$, by the above inequality and the absolute continuity of the integral, we can take δ small enough such that
$2{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx+{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}||\phi |\phantom{\rule{0.2em}{0ex}}dx<\epsilon /3.$
For this δ, since ${u}_{n,k}\to {u}_{k}$ in ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$,
$3{\int }_{{\mathrm{\Omega }}_{\delta }}|{|{u}_{k}|}^{p-1}+{|{u}_{k}|}^{{p}^{\ast }-1}-{|{u}_{n,k}|}^{p-1}-{|{u}_{n,k}|}^{{p}^{\ast }-1}||\phi |\phantom{\rule{0.2em}{0ex}}dx<\epsilon /3$
for n large enough. By (2.4), for this δ, we have
${\int }_{\mathrm{\Omega }\mathrm{\setminus }{\mathrm{\Omega }}_{\delta }}|{|{u}_{n,k}|}^{{p}_{n}-2}{u}_{n,k}\phi -{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}\phi |\phantom{\rule{0.2em}{0ex}}dx<\epsilon /3$

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

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 ${u}_{n,k}^{±}$, we obtain ${\parallel {u}_{n,k}^{±}\parallel }^{p}=\lambda {\parallel {u}_{n,k}^{±}\parallel }_{p}^{p}+{\parallel {u}_{n,k}^{±}\parallel }_{{p}_{n}}^{{p}_{n}}$, where ${u}^{±}=max\left\{±u,0\right\}$. Note that $\lambda <{\lambda }_{1}$, by the Sobolev imbedding theorem, we have $0<\left(1-\frac{\lambda }{{\lambda }_{1}}\right)C\le {\parallel {u}_{n,k}^{±}\parallel }_{{p}^{\ast }}^{{p}_{n}-p}$. It follows that ${u}_{n,k}\to {u}_{k}$ in ${L}^{{p}^{\ast }}\left(\mathrm{\Omega }\right)$ as $n\to \mathrm{\infty }$ for ${u}_{n,k}\to {u}_{k}$ in ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ as $n\to \mathrm{\infty }$. This gives $0<\left(1-\frac{\lambda }{{\lambda }_{1}}\right)C\le {\parallel {u}_{k}^{±}\parallel }_{{p}^{\ast }}^{{p}^{\ast }-p}$, i.e., ${u}_{k}^{±}\ne 0$ for all $k\in \mathbb{N}$. □

Let $\epsilon >0$ and $c\in \mathbb{R}$, we denote
$\begin{array}{c}K:=\left\{u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right):{I}^{\prime }\left(u\right)=0\right\},\phantom{\rule{2em}{0ex}}{K}_{c}:=\left\{u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right):I\left(u\right)=c,{I}^{\prime }\left(u\right)=0\right\},\hfill \\ {K}_{\mu }^{\ast }:=K\mathrm{\setminus }\left(int\left({D}_{\mu }^{+}\right)\cup int\left({D}_{\mu }^{-}\right)\right),\phantom{\rule{2em}{0ex}}{K}_{c,\mu }^{\ast }:={K}_{c}\mathrm{\setminus }\left(int\left({D}_{\mu }^{+}\right)\cup int\left({D}_{\mu }^{-}\right)\right),\hfill \\ {\mathcal{N}}_{c,\mu ,\epsilon }:=\left\{u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right):dist\left(u,{K}_{c}^{\ast }\right)<\epsilon \right\}.\hfill \end{array}$
Thanks to Lemma 2.3, ${u}_{k}\in {K}_{{\mu }_{k}}^{\ast }$ for some ${\mu }_{k}>0$. We claim that $\left\{{u}_{k}\right\}\subset {K}_{\mu }^{\ast }$ for some $\mu >0$. Indeed, if not, then $dist\left({u}_{k},P\right)\to 0$ as $k\to \mathrm{\infty }$ without loss of generality. On the one hand, since ${u}_{k}$ is a solution of (1.1), ${〈{I}^{\prime }\left({u}_{k}\right),{u}_{k}-{S}_{\lambda }\left({u}_{k}\right)〉}_{{W}_{0}^{1,p}\left(\mathrm{\Omega }\right),{W}_{0}^{-1,p}\left(\mathrm{\Omega }\right)}=0$, where ${S}_{\lambda }\left({u}_{k}\right):{\left(-{\mathrm{\Delta }}_{p}\right)}^{-1}\left(\lambda {|{u}_{k}|}^{p-2}{u}_{k}+{|{u}_{k}|}^{{p}^{\ast }-2}{u}_{k}\right)$. On the other hand, by [[17], Lemma 3.7], we have
${〈{I}^{\prime }\left({u}_{k}\right),{u}_{k}-{S}_{\lambda }\left({u}_{k}\right)〉}_{{W}_{0}^{1,p}\left(\mathrm{\Omega }\right),{W}_{0}^{-1,p}\left(\mathrm{\Omega }\right)}\ge C{\parallel {u}_{k}-{S}_{\lambda }\left({u}_{k}\right)\parallel }^{2}{\left(\parallel {u}_{k}\parallel +\parallel {S}_{\lambda }\left({u}_{k}\right)\parallel \right)}^{p-2}$
for $1 and
${〈{I}^{\prime }\left({u}_{k}\right),{u}_{k}-{S}_{\lambda }\left({u}_{k}\right)〉}_{{W}_{0}^{1,p}\left(\mathrm{\Omega }\right),{W}_{0}^{-1,p}\left(\mathrm{\Omega }\right)}\ge C{\parallel {u}_{k}-{S}_{\lambda }\left({u}_{k}\right)\parallel }^{p}$
for $p\ge 2$. Note that by a similar proof of [[14], Lemma 3.3], we can see that ${S}_{\lambda }\left({D}_{\mu }^{±}\right)\subset int\left({D}_{\mu }^{±}\right)$ for μ small enough. Thus, $\parallel {u}_{k}-{S}_{\lambda }\left({u}_{k}\right)\parallel >0$ for k large enough. This implies
${〈{I}^{\prime }\left({u}_{k}\right),{u}_{k}-{S}_{\lambda }\left({u}_{k}\right)〉}_{{W}_{0}^{1,p}\left(\mathrm{\Omega }\right),{W}_{0}^{-1,p}\left(\mathrm{\Omega }\right)}\ge {C}_{k}>0$
for k large enough, which contradicts ${〈{I}^{\prime }\left({u}_{k}\right),{u}_{k}-{S}_{\lambda }\left({u}_{k}\right)〉}_{{W}_{0}^{1,p}\left(\mathrm{\Omega }\right),{W}_{0}^{-1,p}\left(\mathrm{\Omega }\right)}=0$. 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 ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ (cf. [[9], Theorem 1.2]). It follows from [[19], Proposition 7.5] that there exists $\epsilon >0$ such that
$gen\left({\mathcal{N}}_{c,2\epsilon }\right)=gen\left({K}_{c}^{\ast }\right)<+\mathrm{\infty }.$
(2.5)

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

Lemma 2.4 Assume that there exists $\delta >0$ such that ${K}^{\ast }\cap {J}^{c+\delta }\mathrm{\setminus }int\left({J}^{c-\delta }\right)={K}_{c}^{\ast }$ for n large. Then there exists $\gamma >0$ such that $\parallel {I}_{n}^{\prime }\left(u\right)\parallel \ge \gamma$ 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 $\left\{{v}_{n,k}\right\}\subset {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ such that ${lim}_{k\to \mathrm{\infty }}{I}_{n}^{\prime }\left({v}_{n,k}\right)=0$. It is clear that ${I}_{n}$ satisfies the (PS) condition for every $n\in \mathbb{N}$. Hence there exists ${v}_{n}\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ such that, up to a subsequence, ${v}_{n,k}\to {v}_{n}$ in ${W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ as $k\to \mathrm{\infty }$ with ${I}_{n}^{\prime }\left({v}_{n}\right)=0$ and ${I}_{n}\left({v}_{n}\right)\in \left[c-\delta ,c+\delta \right]$. This implies
$c+\delta \ge {I}_{n}\left({v}_{n}\right)=\left(\frac{1}{p}-\frac{1}{{p}_{n}}\right)\left(1-\frac{\lambda }{{\lambda }_{1}}\right){\parallel {v}_{n}\parallel }^{p}\ge \left(\frac{1}{p}-\frac{1}{{p}_{1}}\right)\left(1-\frac{\lambda }{{\lambda }_{1}}\right){\parallel {v}_{n}\parallel }^{p}.$

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

Lemma 2.5 Assume that there exists $\gamma >0$ such that $\parallel {I}_{n}^{\prime }\left(u\right)\parallel \ge \gamma$ 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. It is clear that there exists $L>0$ such that
(2.6)
where
Let ${T}_{n,\lambda }:{W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\mathrm{\setminus }K\to {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$ be the local Lipschitz continuous operator obtained in [[14], Lemma 2.1] and let ${\varphi }_{u}\left(t\right)$ be the solution of the following O.D.E.
$\left\{\begin{array}{c}\frac{d\varphi }{dt}=-\varphi +{T}_{n,\lambda }\left(\varphi \right),\hfill \\ \varphi =u\in {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\mathrm{\setminus }K.\hfill \end{array}$

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

Claim 1: ${\varphi }_{u}\left(t\right)$ 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 $\delta >0$, ${\varphi }_{u}\left(t\right)$ will enter ${\mathcal{N}}_{c,\epsilon }$ before it enters ${\mathcal{Q}}_{n}^{c-\delta }$. Since $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }\subset {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)\mathrm{\setminus }{\mathcal{N}}_{c,2\epsilon }$, there exist $0\le {t}_{1}<{t}_{2}<\tau \left(u\right)$ such that ${\varphi }_{u}\left(t\right)\in {\mathcal{N}}_{c,2\epsilon }\mathrm{\setminus }{\mathcal{N}}_{c,\epsilon }$ for $t\in \left({t}_{1},{t}_{2}\right]$ and
$dist\left({\varphi }_{u}\left({t}_{1}\right),{K}_{c}^{\ast }\right)=2\epsilon ,\phantom{\rule{2em}{0ex}}dist\left({\varphi }_{u}\left({t}_{2}\right),{K}_{c}^{\ast }\right)=\epsilon .$
By [[14], Lemma 2.1], $C{\parallel u-{S}_{n,\lambda }\left(u\right)\parallel }^{2}{\left(\parallel u\parallel +\parallel {S}_{n,\lambda }\left(u\right)\parallel \right)}^{p-2}\le 〈{I}_{n}\left(u\right),u-{T}_{n,\lambda }\left(u\right)〉$. On the other hand, by the choice of ${t}_{1}$ and ${t}_{2}$, we know that ${\varphi }_{u}\left(t\right)\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ for $t\in \left({t}_{1},{t}_{2}\right]$. Thanks to [[17], Lemma 3.8], $\parallel u-{S}_{n,\lambda }\left(u\right)\parallel \ge {\left(\frac{\gamma }{C}\right)}^{1/\left(p-1\right)}$ for large n. This, together with (2.6) and [[14], Lemma 2.1], implies
$\begin{array}{rl}\epsilon & \le \parallel {\varphi }_{u}\left({t}_{2}\right)-{\varphi }_{u}\left({t}_{1}\right)\parallel \le {\int }_{{t}_{1}}^{{t}_{2}}\parallel {\varphi }_{u}\left(t\right)-{T}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel \phantom{\rule{0.2em}{0ex}}dt\\ \le C{\int }_{{t}_{1}}^{{t}_{2}}\parallel {\varphi }_{u}\left(t\right)-{S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel \phantom{\rule{0.2em}{0ex}}dt\\ \le C{\int }_{{t}_{1}}^{{t}_{2}}{\parallel {\varphi }_{u}\left(t\right)-{S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel }^{2}{\left(\parallel {\varphi }_{u}\left(t\right)\parallel +\parallel {S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel \right)}^{p-2}\phantom{\rule{0.2em}{0ex}}dt\\ \le C{\int }_{{t}_{1}}^{{t}_{2}}〈{I}_{n}\left({\varphi }_{u}\left(t\right)\right),{\varphi }_{u}\left(t\right)-{T}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)〉\phantom{\rule{0.2em}{0ex}}dt\\ =C\left({I}_{n}\left({t}_{1}\right)-{I}_{n}\left({t}_{2}\right)\right)\le 4C\delta .\end{array}$

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

Claim 2: There exists ${\tau }_{1}\left(t\right)<\tau \left(u\right)$ such that ${\varphi }_{u}\left({\tau }_{1}\left(u\right)\right)\in {\mathcal{Q}}_{n}^{c-\delta }$ for large n and $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$.

If the claim is not true, then ${\varphi }_{u}\left(t\right)\in {\mathcal{Q}}_{n}^{c+\delta }\mathrm{\setminus }{\mathcal{Q}}_{n}^{c-\delta }$ for all $t\in \left(0,\tau \left(u\right)\right)$. We first consider the case of $\tau \left(u\right)<+\mathrm{\infty }$. In fact, by Claim 1, ${\varphi }_{u}\left(t\right)\notin {\mathcal{N}}_{c,\epsilon }$, i.e., ${\varphi }_{u}\left(t\right)\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ for all $t\in \left(0,\tau \left(u\right)\right)$. Since $\parallel {I}_{n}^{\prime }\left(u\right)\parallel \ge \gamma >0$ for $u\in {\mathcal{A}}_{n,\epsilon }^{c,\delta }$ and large n, we must have
(2.7)
On the other hand, by [[14], Lemma 2.1] and [[17], Lemma 5.2], we have
$\begin{array}{rl}\parallel {\varphi }_{u}\left(t\right)-{\varphi }_{u}\left(0\right)\parallel & \le {\int }_{0}^{t}\parallel {\varphi }_{u}\left(s\right)-{T}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel \phantom{\rule{0.2em}{0ex}}ds\\ \le C{\int }_{0}^{t}\parallel {\varphi }_{u}\left(s\right)-{S}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel \phantom{\rule{0.2em}{0ex}}ds\\ \le C{\int }_{0}^{t}{\left(1+\parallel {\varphi }_{u}\left(s\right)-{S}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel \right)}^{p}\phantom{\rule{0.2em}{0ex}}ds\\ \le C{\int }_{0}^{t}{\left(1+\parallel {\varphi }_{u}\left(s\right)-{S}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel \right)}^{2}{\left(\parallel {\varphi }_{u}\left(s\right)\parallel +\parallel {S}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel \right)}^{p-2}\phantom{\rule{0.2em}{0ex}}ds\\ \le C{\int }_{0}^{t}{\parallel {\varphi }_{u}\left(s\right)-{S}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel }^{2}{\left(\parallel {\varphi }_{u}\left(s\right)\parallel +\parallel {S}_{\lambda ,n}\left({\varphi }_{u}\left(s\right)\right)\parallel \right)}^{p-2}\phantom{\rule{0.2em}{0ex}}ds\\ \le C\left({I}_{n}\left({\varphi }_{u}\left(0\right)\right)-{I}_{n}\left({\varphi }_{u}\left(t\right)\right)\right)\le C.\end{array}$
This means $\parallel {\varphi }_{u}\left(t\right)\parallel \le \parallel u\parallel +C$ for all $t\in \left(0,\tau \left(u\right)\right)$, which contradicts with (2.7). It follows that there must exist ${\tau }_{1}\left(u\right)<\tau \left(u\right)$ such that ${\varphi }_{u}\left({\tau }_{1}\left(u\right)\right)\in {\mathcal{Q}}_{n}^{c-\delta }$ for $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$, large n and $\tau \left(u\right)<+\mathrm{\infty }$. Next, we consider the case of $\tau \left(u\right)=+\mathrm{\infty }$. Since $\parallel u-{S}_{n,\lambda }\left(u\right)\parallel \ge {\left(\frac{\gamma }{C}\right)}^{1/\left(p-1\right)}$ 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
$\begin{array}{rl}\frac{d{I}_{n}\left({\varphi }_{u}\left(t\right)\right)}{dt}& =〈{I}_{n}\left({\varphi }_{u}\left(t\right)\right),-{\varphi }_{u}\left(t\right)+{T}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)〉\\ \le -C{\parallel {\varphi }_{u}\left(t\right)-{S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel }^{2}{\left(\parallel {\varphi }_{u}\left(t\right)\parallel +\parallel {S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel \right)}^{p-2}\\ \le -C{\parallel {\varphi }_{u}\left(t\right)-{S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel }^{2}{\left(1+\parallel {\varphi }_{u}\left(t\right)-{S}_{n,\lambda }\left({\varphi }_{u}\left(t\right)\right)\parallel \right)}^{p-2}\\ \le -C<0.\end{array}$

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

Let
${\eta }_{n}\left(u\right)=\left\{\begin{array}{cc}{\varphi }_{u}\left({\tau }_{1}\left(u\right)\right),\hfill & u\in {\mathcal{A}}_{n,2\epsilon }^{c},\hfill \\ u,\hfill & u\in {\mathcal{Q}}_{n}^{c-\delta }.\hfill \end{array}$

Then, by the continuity of ${\varphi }_{u}\left(t\right)$, ${\eta }_{n}\left(u\right)$ is continuous. Note that ${\varphi }_{u}\left(t\right)$ is odd and ${\tau }_{1}\left(u\right)$ is even, we see that ${\eta }_{n}\left(u\right)$ 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 $\lambda \in \left(0,{\lambda }_{1}\right)$. 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 $\left({K}^{\ast }\cap {J}^{{d}_{\ast }+\delta }\mathrm{\setminus }{J}^{{d}_{\ast }-\delta }\right)\mathrm{\setminus }{K}_{{d}_{\ast }}^{\ast }\ne \mathrm{\varnothing }$ 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 $\left({K}^{\ast }\cap {J}^{{d}_{\ast }+\delta }\mathrm{\setminus }{J}^{{d}_{\ast }-\delta }\right)={K}_{{d}_{\ast }}^{\ast }$ for $\delta <{\delta }_{0}$. Thanks to Lemmas 2.4 and 2.5, there exists ${\eta }_{n}$ such that ${\eta }_{n}\left({\mathcal{A}}_{n,2\epsilon }^{{d}_{\ast }}\cup {\mathcal{Q}}_{n}^{{d}_{\ast }-\delta }\right)\subset {\mathcal{Q}}_{n}^{{d}_{\ast }-\delta }$ 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 $\delta \in \left(0,1\right)$. By the definition of ${d}_{n,k+l}$, there exists $Z\in {\mathrm{\Gamma }}_{k+l}$ such that ${sup}_{Z}{I}_{n}\left(u\right)<{d}_{\ast }+\delta$, where $Z=h\left({B}_{m}\mathrm{\setminus }B\right)\mathrm{\setminus }{D}_{\mu }$, $h\in {G}_{m}$ and $gen\left(B\right)\le m-k-l$. It follows that $h\left({B}_{m}\mathrm{\setminus }B\right)\mathrm{\setminus }{\mathcal{N}}_{{d}_{\ast },2\epsilon }\subset {\mathcal{A}}_{n,2\epsilon }^{c,\delta }\cup {\mathcal{Q}}_{n}^{c-\delta }$. Thus, ${\eta }_{n}\left(h\left({B}_{m}\mathrm{\setminus }B\right)\mathrm{\setminus }{\mathcal{N}}_{{d}_{\ast },2\epsilon }\right)\subset {\mathcal{Q}}_{n}^{{d}_{\ast }-\delta }$. By the choice of δ and ${B}_{m}$, we have ${\eta }_{n}\circ h\in {G}_{m}$. If $gen\left(B\cup {h}^{-1}\left({\mathcal{N}}_{{d}_{\ast },2\epsilon }\right)\right)\le m-k$, then we have
${d}_{\ast }-\delta <{d}_{n,k}\le \underset{{\eta }_{n}\circ h\left({B}_{m}\mathrm{\setminus }\left(B\cup {h}^{-1}\left({\mathcal{N}}_{{d}_{\ast },2\epsilon }\right)\right)\right)}{sup}{I}_{n}\left(u\right)\le {d}_{\ast }-\delta .$
A contradiction. By the properties of gen, we have
$m-k+1\le gen\left(B\cup {h}^{-1}\left({\mathcal{N}}_{{d}_{\ast },2\epsilon }\right)\right)\le gen\left(B\right)+gen\left({\mathcal{N}}_{{d}_{\ast },2\epsilon }\right)\le m-k-l+gen\left({\mathcal{N}}_{{d}_{\ast },2\epsilon }\right).$

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

## Declarations

### Acknowledgements

The work was supported by the Natural Science Foundation of China (11071180, 11171247) and College Postgraduate Research and Innovation Project of Jiangsu Province (CXZZ110082).

## Authors’ Affiliations

(1)
Department of Mathematics, Soochow University

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