\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 220, 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-2017/220\hfil Fractional elliptic equations ] {Multiple solutions of nonlinear fractional elliptic equations via Morse theory} \author[W. Qi, L. Zhao, X. Yan \hfil EJDE-2017/220\hfilneg] {Wei Qi, Lin Zhao, Xingjie Yan} \address{Wei Qi \newline School of physical science and technology, Xuzhou, Jiangsu 221008, China} \email{qiw04@cumt.edu.cn} \address{Lin Zhao (corresponding author)\newline Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China} \email{zhaolinmath@yeah.net} \address{Xingjie Yan \newline Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China} \email{yanxj04@163.com} \dedicatory{Communicated by Zhaosheng Feng} \thanks{Submitted June 4, 2016. Published September 16, 2017.} \subjclass[2010]{35J60, 35B50} \keywords{Fractional elliptic equation; critical point; Morse theory; \hfill\break\indent critical group; sub-supersolutions} \begin{abstract} This article concerns the existence and multiplicity of weak solutions of the nonlinear fractional elliptic problem. We extend some well known results of semilinear Laplacian equations to the nonlocal fractional setting. Using the variational methods based on the critical point theory, sub-supersolutions methods and Morse theory, we show that the problem has at least 6 nontrivial solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In the recent years, there has been a considerable interest to study the partial differential equations involving nonlocal operators. Nonlocal operators often appear in complex systems, such as anomalous diffusion and geophysical flows \cite{Ap,cv, MS2012, MK2004}, a thin obstacle problem \cite{sil}, finance \cite{29} and stratified materials \cite{sv9} etc. A special and important nonlocal operator is the fractional Laplacian operator arising in non-Gaussian stochastic systems. For a stochastic differential system with a $s$-stable L\'evy motion (a non-Gaussian stochastic process) $L_t^s$ for $s \in (0, 1)$ $$ dX_t =b(X_t)dt + dL^s_t, \quad X_0 =x, $$ the corresponding Fokker-Planck equation contains the fractional Laplacian operator $(-\Delta)^s$. When the drift term $b$ in the above stochastic differential system depends on the probability distribution of the system state, the Fokker-Planck equation becomes a nonlinear and nonlocal partial differential equation \cite{Ap}. There are many works about the modeling techniques, well-posedness and regularity of solutions for the nonlocal partial differential equations with the fractional Laplacian operator $(-\Delta)^s$, $s \in (0, 1)$, see \cite{cr,cv,ck,CMN2012,dg,MS2012,npv,sv13,sil}. Motivated by an evident and increasing interest in the current literature on fractional elliptic problems, in this paper, we are interested in the multiplicity of solution of the problem \begin{equation}\label{e1.1} \begin{gathered} (-\Delta)^s u=f(u),\quad x\in\Omega, \\ u=0,\quad x\in\mathcal{C}\Omega, \end{gathered} \end{equation} where $ \mathcal{C}\Omega=\mathbb{R}^n\backslash\Omega$. Note that the boundary condition is given in $\mathbb{R}^n\backslash\Omega$, not simply on $\partial\Omega$, as it appears in the classical case of the Laplacian equation, consistently with the nonlocal character of the fractional Laplacian operator. The hypotheses on the nonlinearity $f(t)$ of problem \eqref{e1.1} are as follows \begin{itemize} \item[(H1)] $f(t)$ is $\mathcal{C}^1$ and $f(0)=0$; \item[(H2)] for all $t\in\mathbb{R}$, $|f'(t)|\leq C(1+|t|^{r-2})$, with $C>0$ a constant and $22s$ has at least six nontrivial solutions: $u_1$ and $u_3$ have the Morse index 0, $u_2$ and $u_4$ have the Morse index 1, and $u_5$ has the Morse index $d\geq2$. \end{theorem} It is worth mentioning that if we choose suitable fractional Sobolev space and the fractional Laplacian operator is replaced by the more general nonlocal operator $\mathcal{L}_K$, $$ \mathcal{L}_Ku(x)=\frac12\int_{\mathbb{R}^n}(u(x+y)+u(x-y)-2u(x))K(y)dy,\quad x\in \mathbb{R}^n, $$ where $K:\mathbb{R}^n\setminus\{0\}\to(0,+\infty)$ is a function such that $$ mK\in L^1(\mathbb{R}^n),\quad m(x)=\min\{|x|^2,1\}. $$ There exists $\theta>0$ and $s\in(0,1)$ such that $K(x)\geq\theta|x|^{-(n+2s)}$ for any $x\in\mathbb{R}^n\setminus\{0\}$ and for any $x\in\mathbb{R}^n\setminus\{0\}$, $$ K(x)=K(-x), $$ then the result in this article also holds. The rest of this article is organized as follows. In Section 2, we present some necessary preliminary notations and results. Section 3 is devoted to the proof of our main result. \section{Preliminaries} In this section, we present some notation and useful results. The classical fractional Sobolev space is $$ H^s(\Omega)=\Big\{u\in L^2(\Omega):\frac{|u(x)-u(y)|}{|x-y|^{\frac{n+2s}{2}}} \in L^2(\Omega\times\Omega)\Big\} $$ with the Gagliardo norm $$ \|u\|_{H^s(\Omega)} =|u|_2+\Big(\int_{\Omega\times\Omega}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy \Big)^{\frac{1}{2}}. $$ For the details about fractional Sobolev spaces, we refer to \cite{npv}. Define $$ H^s_0(\Omega)=\{g\in H^s(\mathbb{R}^n): g=0 \text{ a.e. in } \mathcal{C}\Omega\}. $$ The space $H^s_0(\Omega)$ is not empty and is a Hilbert space with the scalar product. $$ \langle u,v\rangle_{H^s_0(\Omega)}=\int_{\mathbb{R}^n\times\mathbb{R}^n} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy. $$ Note that the above integral can be extended to $\mathbb{R}^n$, since the function belongs to $H^s_0(\Omega)$. Define $\delta:\overline{\Omega}\to \mathbb{R}_+$ by $\delta(x)=\operatorname{dist}(x,\mathbb{R}^N\setminus\Omega)$, $x\in\overline{\Omega}$, and the space $$ C^0_\delta(\overline{\Omega}) =\Big\{u\in C^0(\overline{\Omega})\Big|\frac{u}{\delta^s} \text{admits a continuous extension to } \overline{\Omega} \Big\} $$ with the norm $\|u\|_{0,\delta}=\|\frac{u}{\delta^s}\|_\infty$. The Banach space $C(\Omega)$ is an ordered Banach space with the positive cone $$ C_+=\{u\in C(\overline{\Omega}): u(x)\geq0 \text{ for }x\in\overline{\Omega}\}. $$ The cone has a nonempty interior, given by $$ \operatorname{int}C_+=\{u\in C_+: u(x)>0\text{ for }x\in\Omega\}. $$ We recall the definition of the fractional Laplacian operator. \begin{definition}\rm For $u\in C^\infty_0(\mathbb{R}^n)$ and $s\in(0,1)$, we define $$ (-\Delta)^{s} u=C(n,s)\operatorname{P.V.}\int_{\mathbb{R}^n} \frac{u(x)-u(y)}{|x-y|^{n+2s}}dy, $$ where the principle value (P.V.) is taken as the limit of the integral over $\mathbb{R}^n\backslash B_\epsilon(x)$ as $\epsilon\to0$, with $B_\epsilon(x)$ the ball of radius $\epsilon$ centered at $x$, and $$ C(n,s)=\frac{2s}{2^{1-2s}\pi^{\frac{n}{2}}} \frac{\Gamma(\frac{n+2s}{2})}{\Gamma(1-s)}. $$ Here $\Gamma(\lambda)=\int^\infty_0t^{\lambda-1}e^{-t}dt$ defined for $\lambda>0$. For more information of the fractional Laplacian operator, we refer to \cite{ck,sv13}. \end{definition} Next we give some properties of the nonlocal operator $(-\Delta)^{s}$, which can explain why $(-\Delta)^{s}$ possesses the elliptic property. For the sake of convenience, we assume $C(n,s)=1$. \begin{lemma} \label{lem2.1} The operator $(-\Delta)^{s}$ admits the following properties: \begin{itemize} \item[(i)] If $u$ is a constant, then $(-\Delta)^{s}u=0$. \item[(ii)] Let $x\in \mathbb{R}^n$. For any maximal point $x_0$ such that satisfies $u(x_0)\geq u(x)$, we have $(-\Delta)^{s}u(x_0)\geq0$. Similarly, if $x^{'}$ is a minimal point such that $u(x^{'})\leq u(x)$, then $(-\Delta)^{s}u(x^{'})\leq0$. \item[(iii)] $(-\Delta)^{s}u$ is a positive semidefine operator, i.e., $((-\Delta)^{s}u,u)\geq0$ in the sense of $L^2(\mathbb{R}^n)$ inner product. \end{itemize} \end{lemma} \begin{proof} By the definition of nonlocal operator $(-\Delta)^{s}$, we see that (i) is trivial. If $x_0$ is the maximum point of $u$ in $\mathbb{R}^n$, i.e., $u(x_0)\geq u(x)$, $x\in\mathbb{R}^n$, then by definition it holds \begin{equation} \label{e2.1} (-\Delta)^{s} u(x_0)=C(n,s)\operatorname{P.V.} \int_{\mathbb{R}^n}\frac{u(x_0)-u(y)}{|x-y|^{n+2s}}dy\geq0, \end{equation} which implies property (ii) because a similar argument can be used for the minimal point. By the inner product we have \begin{equation} \label{e2.2} ((-\Delta)^{s}u,u)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\geq0, \end{equation} which shows property (iii) holds. The proof is complete. \end{proof} \begin{definition} \label{def2.2} \rm We say that $u\in H^s_0(\Omega)$ is a weak supersolution of the problem \eqref{e1.1} if the following inequality holds for all $v\in H^s_0(\Omega)$: $$ \frac{1}{2}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy\geq\int_\Omega fv\,dx. $$ The definition of a weak subsolution can be given in an analogous manner. \end{definition} The energy functional for problem \eqref{e1.1} is $$ J(u)=\frac{1}{2}\int_{\mathbb{R}^n} \int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy -\int_{\Omega}F(u(x))dx, $$ where $$ F(x,t)=\int^t_0f(x,s)ds,\quad x\in\Omega,~t\in \mathbb{R}. $$ It is well known that the function $J$ is well defined and Frech\'et differentiable in $H^s_0(\Omega)$, and $$ \langle J^{'}(u),\varphi\rangle =\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(u(x)-u(y))(\varphi(x) -\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy - \int_{\Omega}f(u(x))\varphi(x)dx, $$ for any $\varphi\in H^s_0(\Omega)$. Thus, the critical points of $J$ are solutions of problem \eqref{e1.1}. To obtain our main result, the following eigenvalue problem plays a crucial role. \begin{equation} \label{e2.3} \begin{gathered} (-\Delta)^su=\lambda u ,\quad x\in \Omega,\\ u=0,\quad x\in \mathcal{C}\Omega. \end{gathered} \end{equation} The weak formulation of the problem \eqref{e2.1} is \begin{equation} \label{e2.4} \begin{gathered} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(u(x)-u(y)) (\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\,dx\,dy =\lambda\int_{\Omega}u(x)\varphi(x)dx,\\ u\in H^s_0(\Omega), \quad \varphi\in H^s_0(\Omega). \end{gathered} \end{equation} We say that $\lambda\in \mathbb{R}$ is an eigenvalue of problem \eqref{e2.1}, if there exists a non-trivial solution $u\in H^s_0(\Omega)$ of problem \eqref{e2.1} for some $\lambda>0$. Any solution will be called an eigenfunction corresponding to the eigenvalue $\lambda$. \begin{lemma}[\cite{sv13}] \label{lem2.2} Let $s\in(0,1)$, $n>2s$, and $\Omega$ be an open bounded set of $\mathbb{R}^n$. Then we have \begin{itemize} \item[(1)] the set of the eigenvalues of \eqref{e2.1} consists of a sequence $\{\lambda_n\}_{n\in \mathbb{N}}$ with \begin{equation} \label{e2.5} 0 < \lambda_1 < \lambda_2 \leq \dots \leq \lambda_j \leq \dots ,\quad \lambda_j \to + \infty \quad \text{as }j \to \infty. \end{equation} \item[(2)] The sequence $\{e_n\}_{n\in \mathbb{N}}$ of eigenfunctions corresponding to $\lambda_n$ is an orthonormal basis of $L^2(\Omega)$ and an orthonormal basis of $H^s_0(\Omega)$. \end{itemize} \end{lemma} By Lemma \ref{lem2.2}, we have the direct sum decomposition $$ H^s_0(\Omega)=H_k\oplus H^\bot_k, $$ where $H_k=\operatorname{span}\{e_1,\dots ,e_k\}$ and $H^\bot_k$ denotes orthogonal completement of $H_k$. \begin{lemma} \label{lem2.3} For all $u\in H_k$, $k\in \mathbb{N}$, we have $$ \|u\|^2_{H^s_0(\Omega)}\leq\lambda_k|u|^2_2. $$ \end{lemma} This is so because $u\in H_k$, we have $u=\sum^k_{i=1}u_ie_i$ with $u_i\in \mathbb{R}$, $i=1,\dots ,k$. Then \begin{equation} \label{e2.6} \|u\|^2_{H^s_0(\Omega)}=\sum^k_{i=1}u_i^2\|e_i\|^2_{H^s_0(\Omega)} =\sum^k_{i=1}\lambda_iu^2_i\leq \lambda_k\sum^k_{i=1}u^2_i=\lambda_k|u|^2_2. \end{equation} \begin{lemma} \label{lem2.4} For all $u\in H^\bot_k$, $k\in \mathbb{N}$, we have \begin{equation} \label{e2.7} \|u\|^2_{H^s_0(\Omega)}\geq\lambda_{k+1}|u|^2_2. \end{equation} \end{lemma} For $u\in H^\bot_k$, $u=\sum^\infty_{i=k}u_ie_i$, it is straightforward to check $$ \|u\|^2_{H^s_0(\Omega)}=\sum^\infty_{i=k}u_i^2\|e_i\|^2_{H^s_0(\Omega)} =\sum^\infty_{i=k}\lambda_iu^2_i\geq \lambda_{k+1}\sum^\infty_{i=k}u^2_i =\lambda_{k+1}|u|^2_2. $$ \begin{lemma}[\cite{npv}] \label{lem2.5} Let $s\in(0,1)$, $n>2s$, and $\Omega$ be an open bounded set of $\mathbb{R}^n$. The embedding $H^s_0(\Omega)\hookrightarrow L^{q}(\Omega)$, $q\leq2^*=\frac{2n}{n-2s}$, is continuous. Moreover, the embedding $H^s_0(\Omega)\hookrightarrow L^{q}(\Omega)$, $q<2^*$, is compact. \end{lemma} \begin{lemma}[Weak maximum principle \cite{AS}] \label{lem2.6} If $u\in H^s_0(\Omega)$ is a weak supersolution of \eqref{e1.1} with $f=0$, then $u\geq0$ a.e. in $\Omega$ and $u$ admits a lower semi-continuous representative in $\Omega$. \end{lemma} \begin{lemma}[Strong maximum principle \cite{AS}] \label{lem2.7} If $u\in H^s_0(\Omega)\setminus\{0\}$ is a weak supersolution of \eqref{e1.1} with $f=0$, then $u>0$ in $\Omega$. \end{lemma} \begin{lemma}[Local minimizer \cite{AS}] \label{lem2.8} Let $\Omega$ be a bounded $\mathcal{C}^{1,1}$ domain, $f$ satisfies {\rm (H1)--(H5)}, and $u_0\in H^s_0(\Omega)$. Then, the following two assertions are equivalent: \begin{itemize} \item[(i)] there exists $\rho>0$ such that $J(u_0+v)\geq J(u_0)$ for all $v\in H^s_0(\Omega)\cap C^0_\delta(\overline{\Omega})$, $\|v\|_{0,\delta}\leq \rho$; \item[(ii)] there exists $\varepsilon>0$ such that $J(u_0+v)\geq J(u_0)$ for all $v\in H^s_0(\Omega)$, $\|v\|_{H^s_0(\Omega)}\leq \varepsilon$. \end{itemize} \end{lemma} Let us recall some basic facts about variational methods and Morse theory \cite{kcc,mw}. Suppose that $(X,Y)$ is a pair of topological spaces with $Y\subset X$. We call $Y\subset X$ a topological pair. Let $Y_2\subset Y_1 \subset X$ and $k\geq 0$ is an integer. We denote the $k$th singular homology group by $H_k(Y_1,Y_2)$ for the pair $(Y_1,Y_2)$ with integer coefficients. The critical groups of $\varphi$ at an isolated critical point $u_0\in X$ with $\varphi(u_0)=c$ are defined by \begin{equation*} C_k(\varphi,u_0)=H_k(\varphi^c\cap U, (\varphi^c\cap U)\setminus\{u_0\}),~~\forall k\geq 0, \end{equation*} where $U$ is a neighborhood of $u_0$ such that $K\cap\varphi^c\cap U=\{u_0\}$ with $K=\{u\in X|\varphi'(u)=0\}$ and $\varphi^c=\{u\in X: \varphi(u)\leq c\}$. If $\varphi$ satisfies the $(P-S)$ condition and the critical values of $\varphi$ are bounded from below by some $a>-\infty$, then the critical groups of $\varphi$ at infinity are given by \cite{BL}: $$ C_k(\varphi,\infty)=H_k(X,\varphi^a),\quad k\geq0, $$ By the deformation lemma, $H_k(X,\varphi^a)$ does not depend on the choice of $a$. We have the Morse inequality \begin{equation*} \sum_{k=0}^\infty M_kt^k=\sum_{k=0}^\infty \beta_kt^k+(1+t)Q(t), \end{equation*} where $Q$ is a formal series with non-negative coefficients in $\mathbb{N}$, $$ M_k=\sum_{\varphi'(u)=0}\text{rank} C_k(\varphi,u),\quad \beta_k=\text{rank} C_k(\varphi,\infty). $$ \begin{lemma}[\cite{kcc}]\label{lem2.9} Suppose that $\varphi\in \mathcal{C}^1(X,\mathbb{R})$ satisfies the $P-S$ condition, and $\varphi$ has only finitely many critical points. If for some $k\in \mathbb{N}$, we have $C_k(\varphi,\infty)\neq0$, then $\varphi$ has a critical point $u$ with $C_k(\varphi,u)\neq0$. \end{lemma} \begin{lemma}[\cite{LL}] \label{lem2.10} Suppose that $\varphi\in \mathcal{C}^2(X,\mathbb{R})$, and $u_0$ is an isolated critical point with the finite Morse index $\mu$ and nullity $\nu$. If $\varphi''(u_0)$ is a Fredholm operator, then $C_k(\varphi,u_0)=0$ for $k\notin[\mu,\mu+\nu]$. Moreover, it holds (1) $C_\mu(\varphi,u_0)\neq0$ implies \begin{equation} C_k(\varphi,u_0)=\begin{cases} G&k=\mu, \\ 0&k\neq \mu. \end{cases} \end{equation} (2) $C_{\mu+\nu}(\varphi,u_0)\neq0$ implies \begin{equation} C_k(\varphi,u_0)= \begin{cases} G&k=\mu+\nu, \\ 0&k\neq \mu+\nu. \end{cases} \end{equation} \end{lemma} \section{Main result} We prove a technical lemma before giving the proof of the main theorem. \begin{lemma} \label{lem3.1} If {\rm (H1)} and {\rm (H3)} hold, then $J$ satisfies the P-S condition. \end{lemma} \begin{proof} We only need to show that $|J(u_n)|\leq C$ and $J'(u_n)\to0$ as $n\to\infty$ implies that $\{u_n\}_{n\in\mathbb{N}}$ is a bounded sequence. Since $H_0^s(\Omega)=H_m\bigoplus H_{m}^\perp$, there exists $$ v_n\in H_m\quad \text{and}\quad \omega_n\in H_m^\perp $$ such that $u_n=v_n+\omega_n$, where $m$ is given by (H3). From $J'(u_n)\to0$ as $n\to\infty$, there exists $N\in\mathbb{N}$ such that $\langle J'(u_n),h\rangle\leq\|h\|$, for $n\geq N$ and $h\in H_0^1(\Omega)$. Set $h=-v_n+\omega_n$. Then we have \begin{equation*} \langle J'(u_n),h\rangle=\langle J'(v_n+\omega_n),-v_n+\omega_n\rangle \leq\|-v_n+\omega_n\|_{H^s_0}=\|v_n+\omega_n\|_{H^s_0}, \end{equation*} and \begin{equation*} \langle J'(v_n+\omega_n),-v_n+\omega_n\rangle=-\|v_n\|^2_{H^s_0}+\|\omega_n\|^2_{H^s_0}-\int_{\Omega}f(v_n+\omega_n)(-v_n+\omega_n)dx. \end{equation*} This implies \begin{equation} \label{e3.1} -\|v_n\|^2_{H^s_0}+\|\omega_n\|^2_{H^s_0} \leq\|v_n+\omega_n\|_{H^s_0}+\int_{\Omega}f(v_n+\omega_n)(-v_n+\omega_n)dx. \end{equation} By (H3), we may find a function $g$ satisfying $f(t)=\alpha t+g(t)$ and $\frac{g(t)}{t}\to0$ as $|t|\to\infty$, where $\alpha\in(\lambda_m,\lambda_{m+1})$. In view of this fact and \eqref{e3.1}, we have \begin{align*} &\|\omega_n\|^2_{H^s_0}-\alpha\int_{\Omega}|\omega_n|^2dx -\|v_n\|^2_{H^s_0}+\alpha\int_{\Omega}|v_n|^2dx\\ &\leq \|v_n+\omega_n\|_{H^s_0}+\int_{\Omega}g(v_n+\omega_n)(-v_n+\omega_n)dx. \end{align*} By \eqref{e2.4} and \eqref{e2.5}, it follows that \begin{equation}\label{e3.2} \begin{aligned} &(1-\frac{\alpha}{\lambda_{m+1}})\|\omega_n\|^2_{H^s_0} +(\frac{\alpha}{\lambda_m}-1)\|v_n\|^2_{H^s_0}\\ &\leq \|v_n+\omega_n\|_{H^s_0} +\int_{\Omega}g(v_n+\omega_n)(-v_n+\omega_n)dx. \end{aligned} \end{equation} Next, we deal with term $\int_{\Omega}g(v_n+\omega_n)(-v_n+\omega_n)dx$ of \eqref{e3.2}. Since the continuity of $g$ and $\frac{g(t)}{t}\to0$ as $|t|\to\infty$, we have \begin{gather*} |\frac{g(t)}{t}|<\varepsilon\quad \text{if }|t|>M,\\ |g(t)|\leq K\quad \text{if } |t|\leq M, \end{gather*} for some constants $M,K>0$. By the Sobolev embedding $H^s_0(\Omega)\hookrightarrow L^p(\Omega)$, $1\leq p< \frac{2n}{n-2s}$ (see lemma \ref{lem2.5}), we get \begin{equation} \label{e3.3} \begin{aligned} &\int_{\Omega}g(v_n+\omega_n)(-v_n+\omega_n)dx\\ &\leq \int_{|u_n|>M}|g(v_n+\omega_n)||-v_n+\omega_n|dx +\int_{|u_n|\leq M}|g(v_n+\omega_n)||-v_n+\omega_n|dx\\ &\leq \varepsilon\int_{|u_n|>M}|v_n+\omega_n||-v_n +\omega_n|dx+K\int_{|u_n|\leq M}|-v_n+\omega_n|dx\\ &\leq \varepsilon C_2^2\|v_n+\omega_n\|_{L^2(\Omega)}\|-v_n +\omega_n\|_{L^2(\Omega)}+K\int_{\Omega}|-v_n+\omega_n|dx\\ &\leq \varepsilon C_2^2\|v_n+\omega_n\|_{H^s_0}\|-v_n +\omega_n\|_{H^s_0}+KC_1\|-v_n+\omega_n\|_{H^s_0}\\ &=\varepsilon C_2^2\|v_n+\omega_n\|^2_{H^s_0}+KC_1\|v_n+\omega_n\|_{H^s_0}. \end{aligned} \end{equation} From \eqref{e3.2} and \eqref{e3.3}, we have \begin{equation} \label{e3.4} (1-\frac{\alpha}{\lambda_{m+1}}-\varepsilon C_2^2)\|\omega_n\|^2_{H^s_0} +(\frac{\alpha}{\lambda_m}-1-\varepsilon C_2^2)\|v_n\|^2_{H^s_0} \leq(1+KC_1)\|v_n+\omega_n\|_{H^s_0}. \end{equation} In view of $\alpha\in(\lambda_m,\lambda_{m+1})$, we may set $\varepsilon$ small enough such that \begin{equation*} 1-\frac{\alpha}{\lambda_{m+1}}-\varepsilon C_2^2>0\quad \text{and}\quad \frac{\alpha}{\lambda_m}-1-\varepsilon C_2^2>0. \end{equation*} It follows that $\{v_n\}$ and $\{\omega_n\}$ are bounded sequences. Thus, $\{u_n\}$ is a bounded sequence. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] We divide the proof into three steps. We only deal with $u\in \operatorname{int}C_+$. The case of $u\in -\operatorname{int}C_+$ is similar. \smallskip \noindent\textbf{Step 1.} Find a positive local minimizer. From (H4), we have $f(t)=\beta t+o(t)$ as $t\to0$ with $t>0$ and $\beta\in(\lambda_1,\lambda_2)$. So it holds \begin{equation} \begin{gathered} (-\Delta)^s u=f(u)=\beta u+o(u)>\lambda_1u+o(u)>\lambda_1\varepsilon e_1, \quad x\in\Omega, \\ u=0,\quad x\in \mathcal{C}\Omega, \end{gathered} \end{equation} where $e_1>0$ is the first eigenfunction of the problem \begin{gather*} (-\Delta)^s u=\lambda u,\quad x\in\Omega,\\ u=0,\quad x\in \mathcal{C}\Omega,\\ \end{gather*} with $\max_{\overline{\Omega}}e_1=1$ for $\varepsilon$ small enough. Then $\underline{u}=\varepsilon e_1$ is a strict positive subsolution of the problem \eqref{e1.1}. We consider the following truncation of $f$, \begin{equation} \hat{f}(t)= \begin{cases} f(\underline{u}) &\text{if }t< \underline{u}, \\ f(t) &\text{if } \underline{u}\leq t\leq\xi_+,\\ f(\xi_+) &\text{if } t>\xi_+, \end{cases} \end{equation} and $\hat{F}(u)=\int_0^u\hat{f}(t)dt$, and $\xi_+$ is given by (H5). Then the $\mathcal{C}^1$ function $\widehat{J}:H_0^s\to\mathbb{R}$ is defined by \begin{align*} \widehat{J}(u) &=\frac{1}{2}\int_Q\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy -\int_{\Omega}\hat{F}(u)dx\\ &= \frac{1}{2}\int_Q|u(x)-u(y)|^2K(x-y)\,dx\,dy-\int_\Omega \hat{F}(u)dx. \end{align*} We claim that if $u$ is a critical point of $\widehat{J}$, then $u\in\text{int}C_+$ and $\underline{u} u\}}f(\underline{u})(\underline{u}-u)^+dx. \end{align*} From the subsolution of \eqref{e1.1}, we have \begin{align*} &\int_Q(u(x)-u(y))[(\underline{u}-u)^+(x)-(\underline{u}-u)^+(y)]K(x-y)\,dx\,dy\\ &\leq \int_{\Omega}f(\underline{u})(\underline{u}-u)^+ dx\\ &= \int_{\{\underline{u}> u\}}f(\underline{u})(\underline{u}-u)^+dx. \end{align*} Moreover, we get \begin{align*} &\int_Q(\underline{u}(x)-\underline{u}(y)) [(\underline{u}-u)^+(x)-(\underline{u}-u)^+(y)]K(x-y)\,dx\,dy\\ &\leq\int_{Q}(u(x)-u(y))[(\underline{u}-u)^+(x)-(\underline{u}-u)^+(y)]K(x-y)\,dx\,dy\\ &\leq\int_{\{\underline{u}>u\}}(u(x)-u(y)) [\underline{u}(x)-u(x)-\underline{u}(y)+u(y)]K(x-y)\,dx\,dy. \end{align*} Thus, we have \begin{align*} &\int_{\{\underline{u}>u\}}(\underline{u}(x)-\underline{u}(y)) [(\underline{u}(x)-u(x)-\underline{u}(y)+u(y)]K(x-y)\,dx\,dy\\ &\leq\int_{\{\underline{u}>u\}}(u(x)-u(y)) [\underline{u}(x)-u(x)-\underline{u}(y)+u(y)]K(x-y)\,dx\,dy, \end{align*} and \begin{align*} &\int_{\{\underline{u}>u\}} [(\underline{u}(x)-\underline{u}(y))^2-(\underline{u}(x) -\underline{u}(y))(u(x)-u(y))]K(x-y)\,dx\,dy\\ &\leq\int_{\{\underline{u}>u\}}[(u(x)-u(y))(\underline{u}(x) -\underline{u}(y))-(u(x)-u(y))^2] K(x-y)\,dx\,dy. \end{align*} From the above inequalities, we obtain \begin{align*} &\int_{\{\underline{u}>u\}}\big[(\underline{u}(x)-\underline{u}(y))^2 -2(u(x)-u(y))(\underline{u}(x)-\underline{u}(y))\\ &\quad +(u(x)-u(y))^2\big] K(x-y)\,dx\,dy\leq0. \end{align*} This implies \[ \int_{\{\underline{u}>u\}}|(\underline{u}(x)-\underline{u}(y) -u(x)+u(y)|^2K(x-y)\,dx\,dy\leq0. \] Then we have $m\{x\in\Omega: \underline{u}(x)> u(x)\}=0$ i.e. $u\geq\underline{u}>0$, a.e. $\Omega$. By the regularity of elliptic equations and the strong maximum principle, we obtain $u\in \text{int}C_+$ and $\underline{u}\xi_+\}=0, $$ i.e. $u\leq\xi_+$ a.e. $\Omega$. Again, by the regularity of elliptic equations and the strong maximum principle or Lemma \ref{lem2.6}, we have $u(x)<\xi_+$ for all $x\in\Omega$. It means that $\xi_+$ is a strict supersolution of \eqref{e1.1}. Set $\bar{u}=\xi_+$. By the definition of $\hat{f}$, $u$ is also a critical point of $J$. Then $\underline{u}$ and $\bar{u}$ are a pair of positive strict sub-supersolutions of the problem \eqref{e1.1}. By Lemmas \ref{lem2.9} and \ref{lem2.10}, we have \begin{equation} C_k(J,u_1)= \begin{cases} G &k=0, \\ 0 &k\neq0, \end{cases} \end{equation} (or see \cite{AS}). Obviously, as in the previous proof, $\xi_+> u_1>\underline{u}>0$ holds for all $x\in\Omega$ and $u_1\in \text{int}C_+$. \smallskip \noindent\textbf{Step 2.} Find a positive mountain pass point. The truncation of $f$ is \begin{equation}\label{e3.8} \widetilde{f}(t)= \begin{cases} f(\underline{u}) &\text{if } t< \underline{u}, \\ f(t) &\text{if }\underline{u}\leq t, \end{cases} \end{equation} and $\widetilde{F}(u)=\int_0^u\widetilde{f}(t)dt$. The functional $\widetilde{J}:H_0^s\to\mathbb{R}$, is defined by \begin{equation*} \widetilde{J}(u)=\frac{1}{2}\int_Q \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy -\int_{\Omega}\widetilde{F}(u)dx. \end{equation*} Obviously, the local minimizer $u_1$ of $\widehat{J}$ is also a local minimizer of $\widetilde{J}$. In view of (H3) and mountain pass lemma (see \cite{kcc}), we have a mountain pass critical point $u_2$ of $\widetilde{J}$. As in the previous proof, we obtain $$ u_2\in\operatorname{int}C_+\quad \text{and}\quad \underline{u}