\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 103, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/103\hfil Fractional $p$-Laplacian problems] {Asymmetric critical fractional $p$-Laplacian problems} \author[L. Huang, Y. Yang \hfil EJDE-2017/103\hfilneg] {Li Huang, Yang Yang} \address{Li Huang \newline School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China} \email{1005596725@qq.com} \address{Yang Yang (corresponding author)\newline School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China} \email{yynjnu@126.com} \dedicatory{Communicated by Marco Squassina} \thanks{Submitted November 9, 2016. Published April 18, 2017.} \subjclass[2010]{35B33, 35J92, 35J20} \keywords{Fractional $p$-Laplacian; critical nonlinearity; asymmetric nonlinearity; \hfill\break\indent linking; $\mathbb{Z}_2$-cohomological index} \begin{abstract} We consider the asymmetric critical fractional $p$-Laplacian problem \begin{gather*} (-\Delta)^s_p u = \lambda |u|^{p-2} u + u^{p^\ast_s - 1}_+,\quad \text{in } \Omega;\\ u = 0, \quad \text{in } \mathbb{R}^N\setminus\Omega; \end{gather*} where $\lambda>0$ is a constant, $p^\ast_s=Np/(N - sp)$ is the fractional critical Sobolev exponent, and $u_+(x)=\max\{u(x),0\}$. This extends a result in the literature for the local case $s = 1$. We prove the theorem based on the concentration compactness principle of the fractional $p$-Laplacian and a linking theorem based on the $\mathbb{Z}_2$-cohomological index. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \newcommand{\norm}[2][]{\|#2\|_{#1}} \newcommand{\pnorm}[2][]{\if #1''|#2|_p \else |#2|_{#1} \fi} \section{Introduction} Beginning with the seminal paper of Ambrosetti and Prodi \cite{MR0320844}, elliptic boundary value problems with asymmetric nonlinearities have been extensively studied (see, e.g., Berger and Podolak \cite{MR0377274}, Kazdan and Warner \cite{MR0477445}, Dancer \cite{MR524624}, Amann and Hess \cite{MR549877}, and the references therein). In particular, Deng \cite{MR1137897}, De Figueiredo and Yang \cite{MR1758880}, Aubin and Wang \cite{MR1831984}, Calanchi and Ruf \cite{MR1938385}, and Zhang et al.\ \cite{MR2112476} have obtained existence and multiplicity results for semilinear Ambrosetti-Prodi type problems with critical nonlinearities using variational methods. And the results for the quasilinear Ambrosetti-Prodi type problems can be found in Perera et al.\ \cite{MR1}. Recently, a lot of attention has been given to the study of the elliptic equations involving the fractional $p$-Laplacian, which is the nonlinear nonlocal operator defined on smooth functions by \[ (-\Delta)_p^su(x)=2\lim_{\epsilon\searrow 0} \int_{\mathbb{R}^N\setminus B_{\epsilon}(x)} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}dy, \] where $p\in(1,+\infty)$, $s\in(0,1)$ and $N>sp$. Some motivation that have led to the study of this kind of operator can be found in Caffarelli \cite{C}. The operator $(-\Delta)_p^s$ leads naturally to the quasilinear problem \begin{gather*} (-\Delta)^s_p u = f(x,u),\quad \text{in } \Omega;\\ u = 0, \quad \text{in } \mathbb{R}^N\setminus\Omega; \end{gather*} where $\Omega$ is a domain in $\mathbb{R}^N$. There is currently a rapidly growing literature on this problem when $\Omega$ is bounded with Lipschitz boundary. In particular, fractional p-eigenvalue problems have been studied in \cite{FP,IS,LL,PSY}, global H\"{o}lder regularity in \cite{IMS,MS2}, existence theory in the critical case in \cite{KSY1,MR2,MR4,MS,MS2}. Motivated by \cite{MR1}, in this article, we consider the asymmetric critical fractional $p$-Laplacian problem \begin{equation} \label{1.1} \begin{gathered} (-\Delta)^s_p u = \lambda |u|^{p-2} u + u^{p^\ast_s - 1}_+,\quad \text{in } \Omega; \\ u = 0, \quad \text{in } \mathbb{R}^N\setminus\Omega; \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, $\lambda>0$ is a constant, $p^\ast_s=Np/(N - sp)$ is the fractional critical Sobolev exponent, and $u_+(x)=\max\{u(x),0\}$. We call that $\lambda\in\mathbb{R}$ is a Dirichlet eigenvalue of $(-\Delta)^s_p$ in $\Omega$ if the problem \begin{equation} \label{1.2} \begin{gathered} (-\Delta)^s_p u = \lambda |u|^{p-2} u, \quad \text{in } \Omega;\\ u = 0, \text{in } \mathbb{R}^N\setminus\Omega; \end{gathered} \end{equation} has a nontrivial weak solution. The first eigenvalue $\lambda_1$ is positive, simple, and has an associated eigenfunction $\varphi_1$ that is positive in $\Omega$. And if $\lambda\geq\lambda_2$ is an eigenvalue, $u$ is a $\lambda$-eigenfunction, then $u$ changes sign in $\Omega$. For problem \eqref{1.1} when $\lambda = \lambda_1$, $t \varphi_1$ is clearly a negative solution for any $t < 0$. So here we focus on the case $\lambda$ is not an eigenvalue of $(-\Delta)^s_p$, and our result is the following. \begin{theorem} \label{Theorem 1.1} Let $1sp$, and $\lambda>0$. Then problem \eqref{1.1} has a nontrivial weak solution in the following cases \begin{itemize} \item[(i)] $N=sp^2$ and $0<\lambda<\lambda_1$; \item[(ii)] $N>sp^2$ and $\lambda$ is not an eigenvalue of $(-\Delta)^s_p$. \end{itemize} \end{theorem} \section{Preliminaries and some known results} Let \[ [u]_{s,p} = \Big(\int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^p}{|x - y|^{N+sp}}\, dx dy\Big)^{1/p} \] be the Gagliardo seminorm of a measurable function $u : \mathbb{R}^N \to \mathbb{R}$, and let \[ W^{s,p}(\mathbb{R}^N) = \{u \in L^p(\mathbb{R}^N) : [u]_{s,p} < \infty\} \] be the fractional Sobolev space endowed with the norm \[ \|u\|_{s,p} = \big(\pnorm{u}^p + [u]_{s,p}^p\big)^{1/p}, \] where $|\cdot|_p$ is the norm in $L^p(\mathbb{R}^N)$. We work in the closed linear subspace \[ W^{s,p}_0(\Omega) = \{u \in W^{s,p}(\mathbb{R}^N) : u = 0 \text{ a.e.\ in } \mathbb{R}^N \setminus \Omega\}, \] equivalently renormed by setting $\|\cdot\| = [\cdot]_{s,p}$, which is a uniformly convex Banach space. The imbedding $W^{s,p}_0(\Omega) \hookrightarrow L^r(\Omega)$ is continuous for $r \in [1,p_s^\ast]$ and compact for $r \in [1,p_s^\ast)$. Weak solutions of problem \eqref{1.1} coincide with critical points of the $C^1$-functional \[ I_\lambda (u) = \frac{1}{p}\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx\,dy -\frac{\lambda}{p}\int_\Omega |u|^p\,dx -\frac{1}{p^\ast_s} \int_\Omega u_+^{p^\ast_s}\,dx, \] for $u \in W^{s,p}_0(\Omega)$. We recall that $I_\lambda$ satisfies the Cerami compactness condition at the level $c \in \mathbb{R}$, or the $(C)_c$ condition for short, if every sequence $\{u_j\} \subset W^{s,p}_0(\Omega)$ such that $I_\lambda(u_j) \to c$ and $(1 + \|u_j\|) I_\lambda'(u_j) \to 0$, called a $(C)_c$ sequence, has a convergent subsequence. Let \begin{equation*} S = \inf_{u \in W^{s,p}_0(\Omega) \setminus \{0\}} \frac{\|u\|^p}{|u|^p_{p^\ast_s}} \end{equation*} be the best constant in the Sobolev inequality. From \cite{BMS}, we know that for $1 < p <\infty$, $0ps$, there exists a minimizer for $S$, and for every minimizer $U$, there exist $x_0\in\mathbb{R}^N$ and a constant sign monotone function $u:\mathbb{R}\to\mathbb{R}$ such that $U(x)=u(|x-x_0|)$. In the following, we shall fix a radially symmetric nonnegative decreasing minimizer $U=U(r)$ for $S$. Multiplying $U$ by a positive constant if necessary, we may assume that \begin{equation} \label{71} (- \Delta)_p^s U = U^{p_s^\ast - 1}. \end{equation} For any $\varepsilon > 0$, the function \begin{equation*} \label{91} U_\varepsilon(x) = \frac{1}{\varepsilon^{(N-sp)/p}}\; U\Big(\frac{|x|}{\varepsilon}\Big) \end{equation*} is also a minimizer for $S$ satisfying \eqref{71}. In \cite[Lemma 2.2]{MR4}, the following asymptotic estimates for $U$ were provided. \begin{lemma}[{\cite[Lemma 2.2]{MR4}}] There exist constants $c_1, c_2 > 0$ and $\theta > 1$ such that for all $r \ge 1$, \begin{equation*} \label{10} \frac{c_1}{r^{(N-sp)/(p-1)}} \le U(r) \le \frac{c_2}{r^{(N-sp)/(p-1)}}, \end{equation*} and \begin{equation*} \label{11} \frac{U(\theta\, r)}{U(r)} \le \frac{1}{2}. \end{equation*} \end{lemma} Assume, without loss of generality, that $0\in\Omega$. For $\varepsilon, \delta > 0$, let \[ m_{\varepsilon,\delta} = \frac{U_\varepsilon(\delta)}{U_\varepsilon(\delta) - U_\varepsilon(\theta \delta)}, \] let \[ g_{\varepsilon,\delta}(t) = \begin{cases} 0, & 0 \le t \le U_\varepsilon(\theta \delta),\\ m_{\varepsilon,\delta}^p\, (t - U_\varepsilon(\theta \delta)), & U_\varepsilon(\theta \delta) \le t \le U_\varepsilon(\delta),\\ t + U_\varepsilon(\delta)\, (m_{\varepsilon,\delta}^{p-1} - 1), & t \ge U_\varepsilon(\delta), \end{cases} \] and let \begin{equation*} %\label{defG} G_{\varepsilon,\delta}(t) = \int_0^t g_{\varepsilon,\delta}'(\tau)^{1/p}\, d\tau = \begin{cases} 0, & 0 \le t \le U_\varepsilon(\theta \delta),\\ m_{\varepsilon,\delta}\, (t - U_\varepsilon(\theta \delta)), & U_\varepsilon(\theta \delta) \le t \le U_\varepsilon(\delta),\\ t, & t \ge U_\varepsilon(\delta). \end{cases} \end{equation*} The functions $g_{\varepsilon,\delta}$ and $G_{\varepsilon,\delta}$ are nondecreasing and absolutely continuous. Consider the radially symmetric non-increasing function \[ u_{\varepsilon,\delta}(r) = G_{\varepsilon,\delta}(U_\varepsilon(r)), \] which satisfies \begin{equation*} %\label{20} u_{\varepsilon,\delta}(r) = \begin{cases} U_\varepsilon(r), & r \le \delta,\\ 0, & r \ge \theta \delta. \end{cases} \end{equation*} We have the following estimates for $u_{\varepsilon,\delta}$ which were proved in \cite[Lemma 2.7]{MR4}. \begin{lemma}[{\cite[Lemma 2.7]{MR4}}] \label{lemma2.7} There exists a constant $C = C(N,p,s) > 0$ such that for any $\varepsilon \le \delta/2$, \begin{gather} \|u_{\varepsilon, \delta}\|^p\le S^{N/sp} +C(\frac{\varepsilon}{\delta})^{(N-sp)/(p-1)}, \nonumber\\ \label{2.9} |u_{\varepsilon, \delta}|^p_p \geq \begin{cases} \frac{1}{C} \varepsilon^{sp}\log(\frac{\delta}{\varepsilon}), & \text{if } N=sp^2,\\ \frac{1}{C} \varepsilon^{sp}, & \text{if } N>sp^2, \end{cases}\\ |u_{\varepsilon, \delta}|^{p^\ast_s}_{p^\ast_s} \ge S^{N/sp}-C(\frac{\varepsilon}{\delta})^{N/(p-1)}, \nonumber\\ \label{40} \frac{\|u_{\varepsilon,\delta}\|^p - \lambda |u_{\varepsilon,\delta}|^p}{\pnorm[p_s^\ast]{u_{\varepsilon,\delta}}^p} \le \begin{cases} S - \frac{\lambda}{C}\; \varepsilon^{sp}\, \log \Big(\frac{\delta}{\varepsilon}\Big) + C \Big(\frac{\varepsilon}{\delta}\Big)^{sp}, & N = sp^2,\\ S - \frac{\lambda}{C}\; \varepsilon^{sp} + C \Big(\frac{\varepsilon}{\delta}\Big)^{(N-sp)/(p-1)}, & N > sp^2. \end{cases} \end{gather} \end{lemma} For $p>1$, and the eigenvalues of problem \eqref{1.2}, we define a non-decreasing sequence $\lambda_k$ by means of the cohomological index. This type of construction was introduced for the p-Laplacian by Perera \cite{MR1998432}. (see also Perera and Szulkin \cite{MR2153141}), and it is slightly different from the traditional one, based on the Krasnoselskii genus (which does not give the additional Morse-theoretical information that we need here). We briefly recall the definition of $Z_2$-cohomological index by Fadell and Rabinowitz \cite{MR57:17677}. Let $W$ be a Banach space and let $\mathcal{A}$ denote the class of symmetric subsets of $W \setminus \{0\}$. For $A \in \mathcal{A}$, let $\overline{A} = A/\mathbb{Z}_22$ be the quotient space of $A$ with each $u$ and $-u$ identified, let $f : \overline{A} \to \mathbb{R}\text{P}^\infty$ be the classifying map of $\overline{A}$, and let $f^\ast : H^\ast(\mathbb{R}\text{P}^\infty) \to H^\ast(\overline{A})$ be the induced homomorphism of the Alexander-Spanier cohomology rings. The cohomological index of $A$ is defined by \[ i(A) = \begin{cases} 0, & \text{if } A = \emptyset,\\ \sup \{m \ge 1 : f^\ast(\omega^{m-1}) \ne 0\}, & \text{if } A \ne \emptyset, \end{cases} \] where $\omega \in H^1(\mathbb{R}\text{P}^\infty)$ is the generator of the polynomial ring $H^\ast(\mathbb{R}\text{P}^\infty) = \mathbb{Z}_22[\omega]$. See Perera et al.\ \cite{MR2640827} for details. So the eigenvalues of problem \eqref{1.2}, coincide with critical values of the functional \[ \Psi(u) = \frac{1}{\pnorm{u}^p}, \quad u \in \mathcal{M} = \{u \in W^{s,p}_0(\Omega) : \norm{u} = 1\}. \] Let $\mathcal{F}$ denote the class of symmetric subsets of $\mathcal{M}$, and set \[ \lambda_k := \inf_{M \in \mathcal{F},\; i(M) \ge k}\, \sup_{u \in M}\, \Psi(u), \quad k \in \mathbb{N}. \] Then $0 < \lambda_1 < \lambda_2 \le \lambda_3 \le \cdots \to + \infty$ is a sequence of eigenvalues of problem \eqref{1.2}, and \begin{equation*} \lambda_k < \lambda_{k+1} \implies i(\Psi^{\lambda_k}) = i(\mathcal{M} \setminus \Psi_{\lambda_{k+1}}) = k, \end{equation*} where \[ \Psi^a = \{u \in \mathcal{M} : \Psi(u) \le a\}, \quad \Psi_a = \{u \in \mathcal{M} : \Psi(u) \ge a\}, \quad a \in \mathbb{R}. \] From \cite[Proposition 3.1]{MR4}, the sublevel set $\Psi^{\lambda_k}$ has a compact symmetric subset $E(\lambda_k)$ of index $k$ that is bounded in $L^\infty(\Omega)$. We may assume without loss of generality that $0 \in \Omega$. Let $\delta_0 = \operatorname(0,\partial\Omega)$, take a smooth function $\eta : [0,\infty) \to [0,1]$ such that $\eta(s) = 0$ for $s \le 3/4$ and $\eta(s) = 1$ for $s \ge 1$, set \[ v_\delta(x) = \eta(\frac{|x|}{\delta}) v(x), \quad v \in E(\lambda_k),\, 0 < \delta \le \frac{\delta_0}{2}, \] and let $E_\delta = \{\pi(v_\delta) : v \in E(\lambda_k)\}$, where $\pi : W^{s,p}_0(\Omega) \setminus \{0\} \to \mathcal{M},\, u \mapsto u/\norm{u}$ is the radial projection onto $\mathcal{M}$. \begin{lemma}[{\cite[Proposition 3.2]{MR4}}] \label{Lemma 2.4} There exists a constant $C=C(N, \Omega, p, s, k)>0$ such that for all sufficiently small $\delta>0$, \begin{enumerate} \item[(i)] $\frac{1}{C}\le|\omega|_q\le C$, for all $\omega\in E_\delta, 1\le q\le\infty$, \item[(ii)] $\sup_{\omega\in E_\delta} I_\lambda (\omega) \le\lambda_k+C\delta^{N-sp}$, \item[(iii)] $E_\delta\cap \Psi_{\lambda_{k+1}} = \emptyset, i(E_\delta) = k$, \item[(iv)] $\operatorname{supp}\omega \cap \operatorname{supp}\pi(u_{\varepsilon,\delta}) = \emptyset$ for all $\omega \in E_\delta$, \item[(v)] $\pi(u_{\varepsilon,\delta}) \notin E_\delta$. \end{enumerate} \end{lemma} We need the following two lemmas for the fractional p-Laplacian. \begin{lemma}[{\cite[P.161]{MR0}}] \label{Lemma 2.2} If $\{u_n\}_{n\in\mathbb{N}}\subset W_0^{s,p}(\Omega)$ is such that $u_n \rightharpoonup u$ in $W_0^{s,p}(\Omega)$, and \begin{equation*} \iint _{\mathbb{R}^{2N}} \frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y)) ((u_n-u)(x)-(u_n-u)(y))}{|x-y|^{N+ps}}\,dx\,dy \to 0, \end{equation*} as $n\to\infty$, then $u_n\to u$ in $W_0^{s,p}(\Omega)$ as $n\to\infty$. \end{lemma} \begin{lemma}[{\cite[Theorem 2.5]{MR2}}] \label{Lemma 2.1} Let $\{u_n\}$ be a bounded sequence in $W^{s,p}_0(\Omega)$, let $|D^s u_n|^p(x):=\int_{\mathbb{R}^N}\frac{|u_n(x)-u_n(y)|^p}{|x-y|^{N+ps}}\,dy,$ for a.e.$x\in\mathbb{R}^N $. Then, up to a subsequence, there exists $u\in W^{s,p}_0(\Omega)$, two Borel regular measures $\mu$ and $\nu$, $\Lambda$ denumerable, $x_j\in\Omega, \nu_j\ge 0, \mu_j\ge 0$ with $\mu_j+\nu_j>0 $, such that \begin{gather*} u_n\rightharpoonup u \quad\text{weakly in} \quad W^{s,p}_0(\Omega), \quad\text{and } u_n\to u \quad \text{strongly in} \quad L^p(\Omega),\\ |D^s u_n|^p \xrightarrow{w^\ast} d\mu, \quad |u_n|^{p^\ast_s}\xrightarrow{w^\ast} d\nu,\\ d\mu\ge|D^s u|^p+ \sum_{j\in\Lambda} \mu_j\delta_{x_j}, \quad \mu_j:=\mu(\{x_j\}),\\ d\nu=|u|^{p^\ast_s}+ \sum_{j\in\Lambda} \nu_j\delta_{x_j}, \quad \nu_j:=\nu(\{x_j\}),\\ \mu_j\ge S \nu^{p/p^\ast_s}_j. \end{gather*} \end{lemma} We will prove Theorems \ref{Theorem 1.1} using the following abstract critical point theorem proved in Yang and Perera \cite[Theorem 2.2]{YaPe2}, which was also used successfully in \cite{PSY1,MR1,MR4}, and generalizes the well-known linking theorem of Rabinowitz \cite{R}. \begin{lemma}[{\cite[Theorem 2.2]{YaPe2}}] \label{Lemma 1.3} Let $W$ be a Banach space, let $S=\{u \in W : \|u\|=1\}$ be the unit sphere in $W$, and let $\pi : W \setminus \{0\} \to S, u\mapsto u/ \|u\|$ be the radial projection onto $S$. Let $I$ be a $C^1$-function on $W$ and let $A_0$ and $B_0$ be disjoint nonempty closed symmetric subsets of $S$ such that \[ i(A_0)=i(S \setminus B_0)<\infty. \] Assume that there exist $R>r>0$ and $v\in S \setminus A_0$ such that \[ \sup I(A) \le \inf I(B),\quad \sup I(X)<\infty, \] where \begin{gather*} A=\{tu:u\in A_0, 0\le t\le R\} \cup \{R\pi((1-t)u+tv): u\in A_0, 0\le t\le 1 \}, \\ B=\{ru:u\in B_0\}, \quad X=\{tu:u\in A, \|u\|=R, 0\le t\le 1 \}. \end{gather*} Let $\Gamma=\{\gamma\in C(X, W): \gamma(X)\text{ is closed and } \gamma|_A=id_A \}$, and set \[ c:=\inf_{\gamma\in\Gamma}\sup_{u\in\gamma(X)} I(u). \] Then \[ \inf I(B)\le c \le \sup I(X), \] in particular, $c$ is finite. If, in addition, $I$ satisfies the $(C)_c$ condition, then $c$ is a critical value of $I$. \end{lemma} \section{Proof of Theorem \ref{Theorem 1.1}} First, we will give our main lemma. \begin{lemma} \label{Lemma 2.3} If $\lambda \neq \lambda_1$, then $I_\lambda$ satisfies the $(C)_c$ condition for all $c<\frac{s}{N}S^{N/sp}$. \end{lemma} \begin{proof} Let $ c<\frac{s}{N}S^{N/sp} $, and let $ \{u_j\} $ be a $ (C)_c $ sequence. First we show that $ \{u_j\} $ is bounded. We have \begin{gather}\label{2.12} \frac{1}{p}\iint_{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^p}{|x-y|^{N+ps}}\,dx\,dy -\frac{\lambda}{p}\int_\Omega |u_j|^p\,dx -\frac{1}{p^\ast_s} \int_\Omega u_{j+}^{p^\ast_s}\,dx =c+o(1), \\ \label{2.13} \begin{aligned} &\iint_{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2} (u_j(x)-u_j(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy \\ &-\lambda \int_\Omega |u_j|^{p-2} u_j v \,dx -\int_\Omega u_{j+}^{p^\ast_s-1} v\,dx \\ &=\frac{o(1)\|v\|}{1+\|u_j\|}. \end{aligned} \end{gather} Taking $v=u_j$ in \eqref{2.13} and combing with \eqref{2.12} gives \begin{equation}\label{2.14} \int_\Omega u_{j+}^{p^\ast_s}\,dx=\frac{N}{s}c+o(1). \end{equation} Taking $v=u_{j+}$ in \eqref{2.13}, and using the equality \begin{equation}\label{u+} |u_+(x)-u_+(y)|^p\leq |u(x)-u(y)|^{p-2}(u(x)-u(y))(u_+(x)-u_+(y)), \end{equation} gives \begin{equation*} %\label{11111} \iint_{\mathbb{R}^{2N}} \frac{|u_{j+}(x)-u_{j+}(y)|^p}{|x-y|^{N+ps}}\,dx\,dy \le \lambda \int_\Omega u_{j+}^p \,dx+\int_\Omega u_{j+}^{p^\ast_s}\,dx+o(1). \end{equation*} So $\{u_{j+}\}$ is bounded in $W^{s,p}_0(\Omega)$. Suppose $\rho_j:=\|u_j\|=(\iint_{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^p}{|x-y|^{N+ps}})^{1/p} \to \infty$ for a renamed subsequence. Then $\tilde{u}_j=\frac{u_j}{\|u_j\|}$ converges to some $\tilde{u}$ weakly in $W^{s,p}_0(\Omega)$, strongly in $L^q(\Omega)$ for $1\le q< p^\ast_s$, and a.e.\ in $\Omega$ for a further subsequence. Since the sequence $\{u_{j+}\}$ is bounded, dividing \eqref{2.12} by $\rho_j^p$ and \eqref{2.13} by $\rho_j^{p-1}$ and passing to the limit then gives \begin{gather*} \lambda \int_\Omega |\tilde{u}|^p\,dx =1, \\ \begin{aligned} &\iint _{\mathbb{R}^{2N}} \frac{|\tilde{u}(x)-\tilde{u}(y)|^{p-2} (\tilde{u}(x)-\tilde{u}(y))(v(x)-v(y))}{|x-y|^{N+ps}}\,dx\,dy\\ &=\lambda \int_\Omega |\tilde{u}|^{p-2}\tilde{u} v \,dx, \quad \forall v\in W^{s,p}_0(\Omega), \end{aligned} \end{gather*} respectively. Moreover, since $\tilde{u}_{j+}=u_{j+}/\rho_j \to 0,\quad \tilde{u}\le 0$ a.e. Hence $\tilde{u}=t\varphi_1$ for some $t<0$ and $\lambda=\lambda_1$, this is a contradiction with assumption. So $\{u_j\}$ is bounded, and for a renamed subsequence, it converges to some $u$ weakly in $W_0^{s,p}(\Omega)$ and $L^{p^*_s}(\Omega)$. Since $\{u_{j+}\}$ is bounded, according to Lemma \ref{Lemma 2.1}, a renamed subsequence of which then converges to some $v\ge 0$ weakly in $W^{s,p}_0(\Omega)$, strongly in $L^q(\Omega)$ for $1\le q < p^\ast_s$ and a.e.\ in $\Omega$, and \begin{equation}\label{2.25} |D^s u_{j+}|^p \xrightarrow{w^\ast} d\mu, \quad |u_{j+}|^{p^\ast_s} \xrightarrow{w^\ast} d\nu, \end{equation} then there exists an at most countable index set $\Lambda$ and points $x_i\in\Omega,\quad i\in\Lambda$, such that \begin{equation} \label{2.26} \begin{gathered} d\mu\ge|D^s v|^p+ \sum_{i\in\Lambda} \mu_i\delta_{x_i}, \quad \mu_i:=\mu(\{x_i\}),\\ d\nu=|v|^{p^\ast_s}+ \sum_{i\in\Lambda} \nu_i\delta_{x_i}, \quad \nu_i:=\nu(\{x_i\}), \end{gathered} \end{equation} where $\mu_i,$ $\nu_i\ge 0$, $\mu_i+\nu_i>0$, and $\mu_i\ge S \nu^{p/p^\ast_s}_i$. Now for any $\rho>0$, let $\varphi_{i,\rho}\in C_c^{\infty}(B_{2\rho}(x_i))$ satisfy \begin{equation*} 0\le\varphi_{i,\rho},\quad \varphi_{i,\rho}|_{B_{\rho}}=1,\quad |\varphi_{i,\rho}|_\infty\le 1,\quad |\nabla \varphi_{i,\rho}|_\infty\le C/\rho. \end{equation*} From \cite[(2.14)]{MR2}, for all $w\in L^{p^*_s}(\mathbb{R}^N)$, \begin{equation}\label{rho} \lim_{\rho\searrow 0}\int_{\mathbb{R}^N}|w|^p|D^s \varphi_{i,\rho}|^pdx=0. \end{equation} Testing equation \eqref{2.13} with $\varphi_{i,\rho} u_{j+}$, which is also bounded in $W_0^{s,p}(\Omega)$, from \eqref{u+}, we obtain \begin{align} &o(1) \\ &=\iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) (\varphi_{i,\rho}(x) u_{j+}(x)-\varphi_{i,\rho}(y) u_{j+}(y))}{|x-y|^{N+ps}}\,dx\,dy \nonumber \\ &\quad -\lambda \int_\Omega |u_j|^{p-2} u_j \varphi_{i,\rho} u_{j+} \,dx -\int_\Omega u_{j+}^{p^\ast_s-1} \varphi_{i,\rho} u_{j+}\,dx \nonumber\\ &=\iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) (u_{j+}(x)- u_{j+}(y))}{|x-y|^{N+ps}} \varphi_{i,\rho}(x) \,dx\,dy \nonumber\\ &\quad -\int_\Omega u_{j+}^{p^\ast_s} \varphi_{i,\rho} \,dx \nonumber\\ &\quad +\iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) u_{j+}(y)(\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+ps}}\,dx\,dy \nonumber\\ &\quad -\lambda \int_\Omega |u_j|^{p-2} u_j \varphi_{i,\rho} u_{j+} \,dx \nonumber\\ &\ge \iint _{\mathbb{R}^{2N}} \frac{|u_{j+}(x)-u_{j+}(y)|^p}{|x-y|^{N+ps}} \varphi_{i,\rho}(x) \,dx\,dy-\int_\Omega u_{j+}^{p^\ast_s} \varphi_{i,\rho} \,dx \nonumber\\ &\quad +\iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) u_{j+}(y)(\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+ps}}\,dx\,dy \nonumber\\ &\quad -\lambda \int_\Omega |u_j|^{p-2} u_j \varphi_{i,\rho} u_{j+} \,dx. \label{2.15} \end{align} By \eqref{2.25}, we have \begin{gather*} \iint_{\mathbb{R}^{2N}}\frac{|u_{j+}(x)-u_{j+}(y)|^p}{|x+y|^{N+sp}} \varphi_{i,\rho}(x)dx\,dy\to\int_{\mathbb{R}^N} \varphi_{i,\rho}d\mu,\\ \int_{\Omega}u_{j+}^{p^*_s}\varphi_{i,\rho}dx\to\int_{\Omega}\varphi_{i,\rho}d\nu,\\ \int_{\Omega}u_{j+}^p\varphi_{i,\rho}dx\to\int_{\Omega}v^p\varphi_{i,\rho}dx. \end{gather*} Moreover, by H\"{o}lder's inequality, we obtain \begin{equation} \begin{aligned} & \big| \iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) u_{j+}(y)(\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+ps}}\,dx\,dy \big| \\ &\le \iint _{\mathbb{R}^{2N}} |\frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) u_{j+}(y) (\varphi_{i,\rho}(x)-\varphi_{i,\rho}(y))}{|x-y|^{N+ps}}|\,dx\,dy \\ &\le \Big(\iint _{\mathbb{R}^{2N}}\Big| \frac{|u_j(x)-u_j(y)|^{p-2} (u_j(x)-u_j(y))}{|x-y|^{\frac{(p-1)(N+ps)}{p}}}\Big|^{p/(p-1)}\,dx\,dy\Big)^{(p-1)/p} \\ &\quad \times\Big(\int_{\mathbb{R}^{N}} |u_{j+}|^p | D^s\varphi_{i,\rho}|^p\,dy \Big)^{1/p}. \end{aligned}\label{2.16} \end{equation} Notice that $|D^s\varphi_{i,\rho}|^p\in L^{\infty}(\mathbb{R}^N)$, since \begin{equation}\label{2.17} \int_{\mathbb{R}^N} \frac{|\varphi_{i,\rho}(x) -\varphi_{i,\rho}(y)|^p}{|x-y|^{N+ps}}\,dy \le \frac{C}{\rho^{p}}\int_{\mathbb{R}^N} \frac{\min\{1, |x-y|^p\}}{|x-y|^{N+ps}}\,dy \le \frac{C}{\rho^{p}}, \end{equation} then \begin{equation*} \limsup_{j\to +\infty} \int_{\mathbb{R}^{N}} |u_{j+}|^p |D^s\varphi_{i,\rho}|^p\,dy = \int_{\mathbb{R}^{N}} v^p |D^s\varphi_{i,\rho}|^p\,dy, \end{equation*} passing to the limit in \eqref{2.15} gives, \begin{equation*} \int_{\mathbb{R}^N} \varphi_{i,\rho} \,d\mu \le \int_\Omega \varphi_{i,\rho} \,d\nu +C(\int_{\mathbb{R}^{N}} v^p |D^s\varphi_{i,\rho}|^p\,dy)^{1/p} +\lambda\int_{\Omega}v^p\varphi_{i,\rho}dx. \end{equation*} Letting $\rho\searrow0$ and using \eqref{rho}, gives $\nu_i\ge\mu_i$, which together with $\mu_i\ge S \nu^{p/p^\ast_s}_i$, then give $\nu_i=0$ or $\nu_i\ge S^{N/sp}$. We claim that $\nu_i\ge S^{N/sp}$ is not possible to hold. Indeed, passing to the limit in \eqref{2.14} and by \eqref{2.25} and \eqref{2.26}, then $\nu_i\le \frac{N}{s} c< S^{N/sp}$. So $\nu_i=0$, $\Lambda$ is empty, and \begin{equation*}\label{2.18} \int_\Omega u^{p^\ast_s}_{j+} \,dx \to \int_\Omega v^{p^\ast_s} \,dx, \end{equation*} then $u_{j+}\to v$ strongly in $L^{p^*_s}(\Omega)$ by uniform convexity. Combining the fact that $u_j$ converges to $u$ weakly in $L^{p^*_s}(\Omega)$, \begin{equation*} \int_\Omega u^{p^\ast_s -1}_{j+} (u_j-u)\,dx\to 0. \end{equation*} Now we have \begin{align*} &\langle I'_\lambda(u_j), (u_j-u)\rangle\\ &= \iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) ((u_j-u)(x)-(u_j-u)(y))}{|x-y|^{N+ps}}\,dx\,dy\\ &\quad -\lambda \int_\Omega |u_j|^{p-2} u_j (u_j-u)\,dx -\int_\Omega u^{p^\ast_s-1}_{j+}(u_j-u)\,dx \to 0. \end{align*} Therefore \begin{equation*} \iint _{\mathbb{R}^{2N}} \frac{|u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) ((u_j-u)(x)-(u_j-u)(y))}{|x-y|^{N+ps}}\,dx\,dy \to 0. \end{equation*} By Lemma \ref{Lemma 2.2}, we obtain $u_j\to u$ in $W^{s,p}_0(\Omega)$. \end{proof} \begin{proof}[Proof of Theorem \ref{Theorem 1.1}] We now give the proof for the case when $\lambda>\lambda_1$ is not one of the eigenvalues. For $0<\lambda<\lambda_1$, the proof is similar and simpler. Fix $\lambda'$ such that $\lambda_k<\lambda'<\lambda<\lambda_{k+1}$, and let $\delta>0$ be so small such that $\lambda_k+C\delta^{N-sp}<\lambda'$, in particular, \begin{equation}\label{3.1} \Psi(\omega)<\lambda',\quad \forall\omega\in E_\delta. \end{equation} Then take $A_0=E_\delta$ and $B_0=\Psi_{\lambda_{k+1}}$, and note that $A_0$ and $B_0$ are disjoint nonempty closed symmetric subsets of $\mathcal{M}$ such that \[ i(A_0)=i(\mathcal{M}\backslash B_0)=k. \] Now, let $0<\varepsilon\le \delta/2$, let $R>r>0$, and let $ v_0=\pi(u_{\varepsilon, \delta})\in \mathcal{M}\backslash E_\delta$, and let $A, B$, and $X$ be as in Lemma \ref{Lemma 1.3}. For $u\in\Psi_{\lambda_{k+1}}$, \begin{equation*} I_\lambda(ru)\ge\frac{1}{p}(1-\frac{\lambda}{\lambda_{k+1}})r^p -\frac{1}{p^\ast_s S^{p^\ast_s/p}}r^{p^\ast_s}. \end{equation*} Since $\lambda<\lambda_{k+1}$, it follows that $\inf I_\lambda(B)>0$ if $r$ is sufficiently small. Next we show $I_\lambda\le 0$ on $A$ if $R$ is sufficiently large. For $\omega\in E_\delta$ and $t\ge0$, \begin{align*} I_\lambda(t\omega) &=\frac{1}{p}\|t\omega\|^p-\frac{\lambda}{p}|t\omega|^p_p -\frac{1}{p^\ast_s}|t\omega_+|^{p^\ast_s}_{p^\ast_s}\\ & \le \frac{t^p}{p}(1-\frac{\lambda}{\Psi(\omega)}) \le 0, \end{align*} by \eqref{3.1}. Now let $\omega\in E_\delta$, $0\le t\le 1$, and set $u=\pi((1-t)\omega+tv_0)$. Clearly, $\|(1-t)\omega+tv_0\|\le 1$, and since the supports of $\omega$ and $v_0$ are disjoint by Lemma \ref{Lemma 2.4}(iv), \begin{gather*} |(1-t)\omega+tv_0|^{p^\ast_s}_{p^\ast_s}=(1-t)^{p^\ast_s} |\omega|^{p^\ast_s}_{p^\ast_s}+t^{p^\ast_s}|v_0|^{p^\ast_s}_{p^\ast_s}, \\ |u|^p_p=\frac{|(1-t)\omega+tv_0|^p_p}{\|(1-t)\omega+tv_0\|^p} \ge\frac{(1-t)^p}{\Psi(\omega)}\ge \frac{(1-t)^p}{\lambda'}. \end{gather*} Since \begin{equation}\label{3.2} |v_0|^{p^\ast_s}_{p^\ast_s} = \frac{|u_{\varepsilon, \delta }|^{p^\ast_s}_{p^\ast_s}}{\|u_{\varepsilon, \delta}\|^{p^\ast_s}}\ge \frac{1}{S^{N/(N-sp)}}+O(\varepsilon^{(N-sp)/(p-1)}), \end{equation} it follows that \begin{equation}\label{3.3} \begin{aligned} |u_+|^{p^\ast_s}_{p^\ast_s} &= \frac{|[(1-t)\omega+tv_0]_+|^{p^\ast_s}_{p^\ast_s}}{\|(1-t) \omega+tv_0\|^{p^\ast_s}} \\ &\ge (1-t)^{p^\ast_s} |\omega_+|^{p^\ast_s}_{p^\ast_s} + t^{p^\ast_s}|v_0|^{p^\ast_s}_{p^\ast_s} \\ &\ge t^{p^\ast_s}|v_0|^{p^\ast_s}_{p^\ast_s} \ge\frac{t^{p^\ast_s}}{C}, \end{aligned} \end{equation} if $\varepsilon$ is sufficiently small, where $C = C(N,\Omega,p,s,k) > 0$. Then \begin{equation}\label{3.4} \begin{aligned} I_\lambda(Ru) &=\frac{R^p}{p}\|u\|^p-\frac{\lambda R^p}{p}|u|^p_p -\frac{R^{p^\ast_s}}{p^\ast_s}|u_+|^{p^\ast_s}_{p^\ast_s} \\ &\le -\frac{R^p}{p}\big[\frac{\lambda}{\lambda'}(1-t)^p-1\big] -\frac{t^{p^\ast_s}}{p^\ast_s C}R^{p^\ast_s}. \end{aligned} \end{equation} The above expression is clearly non-positive if $t\le 1-(\lambda'/\lambda)^{1/p}=:t_0$. For $t>t_0$, it is non-positive if $R$ is sufficiently large. Now, it only remains to show that \begin{equation}\label{3.5} \sup I_\lambda(X)<\frac{s}{N} S^{N/sp}, \end{equation} if $\epsilon$ is sufficiently small, where \begin{equation*} X=\{\rho\pi((1-t)\omega+tv_0):\omega\in E_\delta, 0\le t\le 1, 0\le \rho \le R\}. \end{equation*} Set again $u=\pi((1-t)\omega+tv_0)$. From \eqref{3.4}, $I_\lambda(\rho u)\le 0$, for all $0\le \rho \le R$, if $0\le t \le t_0$. So we only need to consider the case that $1\ge t\ge t_0$. Then \begin{equation}\label{3.6} \begin{split} \sup_{0\le\rho\le R} I_\lambda(\rho u) &\le \sup_{\rho\ge 0}\Big[\frac{\rho^p}{p}(1-\lambda|u|^p_p) -\frac{\rho^{p^\ast_s}}{p^\ast_s}|u_+|^{p^\ast_s}_{p^\ast_s}\Big]\\ &=\frac{s}{N}\Big[\frac{(1-\lambda|u|^p_p)_+}{|u_+|^p_{p^\ast_s}}\Big]^{N/sp}\\ &=\frac{s}{N}\Big[\frac{(\|(1-t)\omega +tv_0\|^p -\lambda|(1-t)\omega+tv_0|^p_p)_+}{|[(1-t)\omega+tv_0]_+|^p_{p^\ast_s}}\Big]^{N/sp}. \end{split} \end{equation} From the arguments in \cite[pp.17-18 (3.15)-(3.17)]{MR4}, \begin{equation}\label{3.9} \norm{(1 - t)\, \omega + tv_0}^p \le \frac{\lambda}{\lambda'}\, (1 - t)^p + t^p + C \varepsilon^{N-(N-sp)\, q/p}, \end{equation} where $q \in( N(p - 1)/(N - sp),p)$, \begin{equation} \label{3.10} \begin{gathered} \pnorm{(1 - t)\,\omega + tv_0}^p = (1 - t)^p \pnorm{\omega}^p + t^p \pnorm{v_0}^p,\\ \pnorm[p_s^\ast]{[(1 - t)\, \omega + tv_0]_+}^{p_s^\ast} \ge (1 - t)^{p_s^\ast} \pnorm[p_s^\ast]{\omega_+}^{p_s^\ast} + t^{p_s^\ast} \pnorm[p_s^\ast]{v_0}^{p_s^\ast}. \end{gathered} \end{equation} By \eqref{3.2}, $\pnorm[p_s^\ast]{v_0}$ is bounded away from zero, if $\varepsilon$ is sufficiently small, so the last expression in \eqref{3.10} is bounded away from a certain number for $1\geq t\ge t_0$. It follows from \eqref{3.9}, \eqref{3.10} and $|\omega|_p\geq\frac{1}{\lambda'}$ by \eqref{3.1}, that \begin{align*} &\frac{\norm{(1 - t)\, \omega + tv_0}^p - \lambda \pnorm{(1 - t) \omega + tv_0}^p}{\pnorm[p_s^\ast]{[(1 - t) \omega + tv_0]_+}^{p}}\\ &\le \frac{1 - \lambda \pnorm{v_0}^p}{\pnorm[p_s^\ast]{v_0}^p} + C \varepsilon^{N-(N-sp) q/p}\\ &\le\frac{\|u_{\varepsilon,\delta}\|^p-\lambda |u_{\varepsilon,\delta}|^p} {\pnorm[p_s^\ast]{u_{\varepsilon,\delta}}^p}+ C \varepsilon^{N-(N-sp)\, q/p}\\ &\le S - (\frac{\lambda}{C} - C \varepsilon^{(N-sp^2)/(p-1)} - C \varepsilon^{(N-sp)(1-q/p)}) \varepsilon^{sp}, \end{align*} by $v_0 = u_{\varepsilon,\delta}/\|u_{\varepsilon,\delta}\|$, and \eqref{40}. Since $N > sp^2$ and $q < p$, it follows from this that the last expression in \eqref{3.6} is strictly less than $\frac{s}{N}\, S^{N/sp}$ if $\varepsilon$ is sufficiently small. So $0