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

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.

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]

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.

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 $\lambda <{\lambda }_1$.

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

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 $n\in \mathbb{N}$.

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 $k\in \mathbb{N}$.

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 $J_n^c:=\{u\in W_0^{1,p}(\mathrm{\Omega }):I_n(u)\le c\}$ 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.

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.

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 $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$.

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 $u\in {\mathcal{A}}_{n,2\epsilon }^{c,\delta }$.