\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 238, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/238\hfil Least energy sign-changing solutions] {Least energy sign-changing solutions for nonlinear problems involving fractional Laplacian} \author[Z. Gao, X. Tang, W. Zhang \hfil EJDE-2016/??\hfilneg] {Zu Gao, Xianhua Tang, Wen Zhang} \address{Zu Gao \newline School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China} \email{gaozu7@163.com} \address{Xianhua Tang \newline School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China} \email{tangxh@mail.csu.edu.cn} \address{Wen Zhang \newline School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China} \email{zwmath2011@163.com} \thanks{Submitted June 21, 2016. Published August 31, 2016.} \subjclass[2010]{35R11, 58E30} \keywords{Nonlinear problems; sign-changing solutions; nonlocal term} \begin{abstract} In this article, we study the existence of least energy sign-changing solutions for nonlinear problems involving fractional Laplacian. By introducing some new ideas and combining constraint variational method with the quantitative deformation lemma, we prove that the problem possesses one least energy sign-changing solution. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} This article concerns the nonlinear problem involving fractional Laplacian \begin{equation}\label{1.1} \begin{gathered} (- \Delta)^{\alpha}u=f(x, u), \quad x\in\Omega,\\ u=0,\quad x\in\mathbb{R}^{N}\backslash\Omega, \end{gathered} \end{equation} where $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with smooth boundary, $0<\alpha<1$, $N>2\alpha$, $(- \Delta)^{\alpha}$ is the fractional Laplacian of order $\alpha$, $f\in{C}(\Omega\times \mathbb{R},\mathbb{R})$. To prove our results, we use the following assumptions: \begin{itemize} \item[(A1)] $\lim_{s\to 0}f(x,s)/s=0$, uniformly in $x\in\Omega$; \item[(A2)] $\lim_{|s|\to \infty}f(x,s)/ s^{2_\alpha^{*}-1}=0$, uniformly in $x\in\Omega$, where $2_{\alpha}^{\ast}=\frac{2N}{N-2\alpha}$; \item[(A3)] $\lim_{|s|\to \infty}f(x,s)/|s|=+\infty$ for a.e. $x\in\Omega$; \item[(A4)] $f(x,s)/|s|$ is increasing in s on $\mathbb{R}\backslash \{0\}$ for every $x\in\Omega$. \end{itemize} In recent years, nonlinear problems involving fractional Laplacian have been investigated extensively. Indeed, they have impressive applications in many fields, such as thin obstacle problem, optimization, finance, phase transitions, anomalous diffusion and so on. For previous related results see \cite{GPE,LC1,LC2,CM,CX, SGE,PAJ,GH,K, NL,SS,S,LS,YR,ZTZ} and the references therein. Precisely, under the assumption that the nonlinearity satisfies the Ambrosetti-Rabinowitz condition or is indeed of perturbative type, the author proved some existence results of solutions for fractional Schr\"{o}dinger equations in \cite{SS}. Using mountain pass theorem, Raffaella and Servadei studied the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions in \cite{SV}. In fact, by the extension theorem in \cite{LC3} Caffarelli and Silvestrein made greatest achievement in overcoming the difficulty, which is the nonlocality of fractional Laplacian $(-\Delta)^{\alpha}$ in the fractional Schr\"{o}dinger equation. Moreover, a great deal of progress has been made to the fractional Laplacian equations after the work \cite{LC3}. We refer to \cite{CXJ,CW,TWY,TJ,TKM,ZXZ} for the existence results and multiplicity results of solutions, and to \cite{XY1,XY2} for the regularity results, maximum principle, uniqueness result and other properties. As we know, a great attention has been devoted to the existence and multiplicity of positive and nodal solutions of elliptic problems in recent years, see for example \cite{BC,BW,CMT,LZ,T1,T2,ZTZ1} and the references therein. Actually, with the descended flow method and harmonic extension techniques, Chang and Wang studied the existence and multiplicity of sign-changing solutions in \cite{CW}. Via costrained minimization method, Tang \cite{T3,T4,T5,LT,LT1} obtained the existence of Nehari-type ground state positive solutions. By combing minimax method with invariant sets of descending flow, some results about nodal solutions have been obtained in \cite{LLS}. Motivated by papers above, and we especially borrow some ideas from \cite{LT}. What is more, we are interested in Problem \eqref{1.1} with constraint variational method and quantitative deformation lemma, and study the existence of a least energy sign-changing solution. For any measurable function $u:\mathbb{R}^{N}\to \mathbb{R}$ with respect to the Gagliardo norm \[ [u]_{\alpha}=\Big(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y\Big)^{1/2}. \] We introduce the fractional Sobolev space \[ H^{\alpha}(\mathbb{R}^{N})=\{u\in L^2({\mathbb{R}}^{N}): [u]_{\alpha}<+\infty\}, \] which is a Hilbert space. A complete introduction to fractional Sobolev spaces can be found in \cite{EGE}. We also define a closed subspace \[ X(\Omega)=\{u\in H^{\alpha}(\mathbb{R}^{N}): u=0\text{ a.e. in }\mathbb{R}^{N} \backslash\Omega\}. \] Then, by \cite{SS}, $X(\Omega)$ is a Hilbert space with the inner product \[ (u,v)=\iint_{\Omega\times\Omega}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y,\quad \forall u,v\in X(\Omega), \] and the corresponding norm $\|\cdot\|_{X}=[\cdot]_{\alpha}$. For $u\in X(\Omega$), set \begin{equation}\label{1.2} \Phi(u)=\frac{1}{2}\|u\|_{X}^2-\int_{\Omega}F(x,u)\mathrm{d}x, \end{equation} where $F(x,u)=\int_{0}^{u}f(x, t)\mathrm{d}t$. Then $\Phi\in C^{1}(X(\Omega),\mathbb{R})$ and \begin{equation}\label{1.3} \langle\Phi'(u),v\rangle=\iint_{\Omega\times\Omega} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{2\alpha +N}}\mathrm{d}x\mathrm{d}y -\int_{\Omega}f(x,u)v\mathrm{d}x, \end{equation} for all $u,v\in X(\Omega)$. Obviously, its critical points are weak solutions of Problem \eqref{1.1}. Furthermore, if $u\in X(\Omega)$ is a solution of \eqref{1.1} with $u^{\pm}\neq0$, then $u$ is a sign-changing solution, where \[ u^{+}(x):=\max\{u(x),0\}\quad \text{and} \quad u^{-}(x)=:\min\{u(x),0\}. \] We set \[ \mathcal{M}:=\{u\in X(\Omega):u^{\pm}\neq0,~ \langle\Phi'(u),u^{+}\rangle =\langle\Phi'(u),u^{-}\rangle=0\}, \] and define \[ m=\inf_{u\in\mathcal{M}}\Phi(u). \] Throughout this paper, $\|\cdot\|_{p}$ denotes the usual norm in $L^{p}(\Omega)$. \begin{theorem} \label{thm1.1} Assume that conditions {\rm (A1)--(A4)} hold. Then \eqref{1.1} possesses one least energy sign-changing solution $u\in \mathcal{M}$ such that $\inf_{u\in\mathcal{M}}\Phi(u)=m>0$. \end{theorem} The rest of this article is organized as follows. In Section $2$, we prove several lemmas, which are crucial to investigate our main result. The proof of Theorem \ref{thm1.1} is given in Section $3$. \section{Preliminary results} \begin{lemma}[{\cite[Lemma 2.1]{TWW}}] \label{lem2.1} For any $a,b\in\mathbb{R}$, we have \begin{itemize} \item[(i)] $(ka)^{\pm}=ka^{\pm}$, for~all $k\geq0 $, $|a^{\pm}-b^{\pm}|\leq |a-b|$; \item[(ii)] $(a-b)(a^{+}-b^{+})\geq(a^{+}-b^{+})^2$ and $(a-b)(a^{-}-b^{-})\geq(a^{-}-b^{-})^2$; \item[(iii)] $(a^{+}-b^{+})(a^{-}-b^{-})\geq0$. \end{itemize} \end{lemma} By simple computations from the above lemma, we obtain the following lemma. \begin{lemma} \label{lem2.2} Under assumptions {\rm (A1)} and {\rm (A2)}, for any $u\in X(\Omega)$, the following facts hold: \begin{itemize} \item[(i)] $ \|u^{\pm}\|_{X}\leq\|u\|_{X}$; \item[(ii)] \begin{align*} (u,u^{\pm})&=(u^{\pm},u^{\pm})-\iint_{\Omega\times\Omega} \frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}}\mathrm{d}x\mathrm{d}y -\iint_{\Omega\times\Omega}\frac{u^{-}(x)u^{+}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y\\ &=(u^{\pm},u^{\pm})-2\iint_{\Omega\times\Omega} \frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}}\mathrm{d}x\mathrm{d}y; \end{align*} \item[(iii)] \begin{align*} \langle\Phi'(u),u^{\pm}\rangle &=\langle\Phi'(u^{\pm}),u^{\pm}\rangle -\iint_{\Omega\times\Omega}\frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y -\iint_{\Omega\times\Omega}\frac{u^{-}(x)u^{+}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y\\ &=\langle\Phi'(u^{\pm}),u^{\pm}\rangle-2\iint_{\Omega\times\Omega} \frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}}\mathrm{d}x\mathrm{d}y. \end{align*} \end{itemize} \end{lemma} In what follows, we denote \[ B(u):=-\iint_{\Omega\times\Omega}\frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y\,. \] It is obvious that $B(u)\geq0$. \begin{lemma} \label{lem2.3} Assume {\rm (A1)} and {\rm (A2)}, and let $\{u_{n}\}$ be a bounded sequence in $X(\Omega)$. Then up to a subsequence, still denoted by $\{u_{n}\}$, there exists $u\in X(\Omega)$ such that \begin{itemize} \item[(i)] \[ \lim_{n\to \infty} \int_{\Omega}|u_{n}^{\pm}|^{p}\mathrm{d}x =\int_{\Omega}|u^{\pm}|^{p}\mathrm{d}x,\quad \forall p\in[2,2_{\alpha}^{*}); \] \item[(ii)] \[ \lim_{n\to \infty}\int_{\Omega}u_{n}f(x,u_{n})\mathrm{d}x=\int_{\Omega}uf(x,u)\mathrm{d}x; \] \item[(iii)] \[ \lim_{n\to \infty}\int_{\Omega}F(x,u_{n})\mathrm{d}x=\int_{\Omega}F(x,u)\mathrm{d}x; \] \item[(iv)] \[ \liminf_{n\to \infty}\langle\Phi'(u_{n}),u_{n}^{\pm}\rangle \geq\langle\Phi'(u),u^{\pm}\rangle. \] \end{itemize} \end{lemma} \begin{proof} (i)--(iii) are easily proved; so we omit their proofs. (iv)From (ii), Fatou's Lemma and (iii) of Lemma \ref{lem2.2}, it follows that \begin{align*} &\langle\Phi'(u),u^{\pm}\rangle \\ &=\langle\Phi'(u^{\pm}),u^{\pm}\rangle -2\iint_{\Omega\times\Omega}\frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y\\ &=\iint_{\Omega\times\Omega}\frac{[u^{\pm}(x)-u^{\pm}(y)]^2}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y -2\iint_{\Omega\times\Omega}\frac{u^{+}(x)u^{-}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y -\int_{\Omega}u^{\pm}f(x,u^{\pm})\mathrm{d}x\\ &\leq\liminf_{n\to \infty}\Big\{\iint_{\Omega\times\Omega}\frac{[u_{n}^{\pm}(x) -u_{n}^{\pm}(y)]^2}{|x-y|^{2\alpha +N}}\mathrm{d}x\mathrm{d}y -2\iint_{\Omega\times\Omega}\frac{u_{n}^{+}(x)u_{n}^{-}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y\Big\}\\ & \quad -\lim_{n\to \infty}\int_{\Omega}u^{\pm}_{n}f(x,u^{\pm}_{n})\mathrm{d}x \\ &=\liminf_{n\to \infty}\langle\Phi'(u_{n}),u_{n}^{\pm}\rangle. \end{align*} This shows that (iv) holds. \end{proof} \begin{lemma} \label{lem2.4} Under assumptions {\rm (A1)} and {\rm (A2)}, if $\{u_{n}\}$ is a bounded sequence in $\mathcal{M}$ and $q\in(2,2_{\alpha}^{*})$, we have \[ \liminf_{n\to \infty}\int_{\Omega}|u_{n}^{\pm}|^{q}dx>0. \] \end{lemma} \begin{proof} From (A1) and (A2), for any $\varepsilon>0$ and fixed $\tau\in [2,2_{\alpha}^{*})$, there exists $C_{\varepsilon}>0$ such that \begin{equation}\label{2.1} |sf(x,s)|\leq\varepsilon|s|^2+C_{\varepsilon}|s|^{\tau} +\varepsilon|s|^{2_{\alpha}^{*}},\quad \forall x\in\Omega,s\in\mathbb{R}. \end{equation} For $u_{n}\in\mathcal{M}$, we have $\langle\Phi'(u_{n}),u_{n}^{\pm}\rangle=0$. From (iii) of Lemma \ref{lem2.2}, we have \[ \langle\Phi'(u_{n}^{\pm}),u_{n}^{\pm}\rangle -2\iint_{\Omega\times\Omega}\frac{u_{n}^{+}(x)u_{n}^{-}(y)}{|x-y|^{2\alpha +N}} \mathrm{d}x\mathrm{d}y=0, \] which, together with Sobolev embedding and \eqref{2.1}, for $q\in(2,2_{\alpha}^{*})$, yields \begin{equation}\label{2.2} \begin{aligned} \|u_{n}^{\pm}\|_{X}^2 &\leq\int_{\Omega}u_{n}^{\pm}f(x,u_{n}^{\pm})\mathrm{d}x \\ &\leq\varepsilon\int_{\Omega}|u_{n}^{\pm}|^2\mathrm{d}x +C_{\varepsilon}\int_{\Omega}|u_{n}^{\pm}|^{q}\mathrm{d}x +\varepsilon\int_{\Omega}|u_{n}^{\pm}|^{2_{\alpha}^{*}}\mathrm{d}x\\ &\leq\varepsilon \gamma^{-2}_2\|u_{n}^{\pm}\|_{X}^2 +C_{\varepsilon}\gamma^{-2}_{q}\|u_{n}^{\pm}\|_{X}^2\|u_{n}^{\pm}\|_{q}^{q-2} +\varepsilon \gamma^{-2_{\alpha}^{*}}_{2_{\alpha}^{*}}\|u_{n}^{\pm} \|_{X}^{2_{\alpha}^{*}}\,, \end{aligned} \end{equation} where $\gamma_{s}:=\inf_{\|u\|_{s}=1}\|u\|_{X}$, $2\leq s\leq 2_{\alpha}^{*}$. From the boundedness of $\{u_{n}\}$, there is $M$ such that \[ \|u_{n}^{\pm}\|_{X}^{2_{\alpha}^{*}-2}\leq M. \] From \eqref{2.2}, taking $\varepsilon=\min \{ \gamma^2_2/4,\gamma^{2_{\alpha}^{*}}_{2_{\alpha}^{*}}/4M\}$, $C_{0}\geq C_{\varepsilon}$, it follows that \[ \frac{1}{2} \leq C_{0}\gamma^{-2}_{q}\|u_{n}^{\pm}\|_{q}^{q-2}\,. \] Then \[ \liminf_{n\to \infty}\int_{\Omega}|u_{n}^{\pm}|^{q}\mathrm{d}x \geq\Big(\frac{\gamma^2_{q}}{2C_{0}}\Big)^{\frac{1}{q-2}}>0. \] \end{proof} \begin{lemma} \label{lem2.5} Under assumptions {\rm (A1), (A2), (A4)}, for any $u\in X(\Omega)$ with $u^{\pm}\neq0$, $s,t\geq0$ and $(s-1)^2+(t-1)^2\neq 0$, we have \[ \Phi(u) > \Phi(su^{+}+tu^{-})+\frac{1-s^2}{2}\langle\Phi'(u),u^{+}\rangle +\frac{1-t^2}{2}\langle\Phi'(u),u^{-}\rangle+B(u)(s-t)^2. \] \end{lemma} \begin{proof} For $\tau\neq0$, (A4) yields \begin{gather*} f(x,s)<\frac{f(x,\tau)}{|\tau|}|s|,\quad |s|<|\tau|; \\ f(x,s)>\frac{f(x,\tau)}{|\tau|}|s|,\quad |s|>|\tau|. \end{gather*} It follows that \[ \frac{1-\theta^2}{2}\tau f(x,\tau) >\int_{\theta\tau}^{\tau} f(x,s)\mathrm{d}s,\quad \forall x\in\Omega,\; \tau\neq0,\; \theta\geq0~ \text{ and } \theta\neq1. \] Thus, we deduce that \begin{align*} &\Phi(u)-\Phi(su^{+}+tu^{-})\\ &=\frac{1-s^2}{2}\langle\Phi'(u),u^{+}\rangle +\frac{1-t^2}{2}\langle\Phi'(u),u^{-}\rangle\\ &\quad +\int_{\Omega}\big[\frac{1-s^2}{2}f(x,u^{+})u^{+} -\left(F(x,u^{+})-F(x,su^{+})\right)\big]\mathrm{d}x \\ &\quad +\int_{\Omega}\big[\frac{1-t^2}{2}f(x,u^{-})u^{-} -\left(F(x,u^{-})-F(x,tu^{-})\right)\big]\mathrm{d}x +B(u)(s-t)^2\\ &=\frac{1-s^2}{2}\langle\Phi'(u),u^{+}\rangle +\frac{1-t^2}{2}\langle\Phi'(u),u^{-}\rangle +\int_{\Omega}\big[\frac{1-s^2}{2}f(x,u^{+})u^{+} \\ &\quad -\int_{su^{+}}^{u^{+}}f(x,\xi)\mathrm{d}\xi\big]\mathrm{d}x +\int_{\Omega}\Big[\frac{1-t^2}{2}f(x,u^{-})u^{-} -\int_{tu^{-}}^{u^{-}}f(x,\xi)\mathrm{d}\xi\Big]\mathrm{d}x \\ &\quad +B(u)(s-t)^2 \\ &>\frac{1-s^2}{2}\langle\Phi'(u),u^{+}\rangle +\frac{1-t^2}{2}\langle\Phi'(u),u^{-}\rangle +B(u)(s-t)^2, \end{align*} for all $s,t\geq 0$, $(s-1)^2+(t-1)^2\neq 0$. \end{proof} From Lemma \ref{lem2.5}, we have the following two corollaries. \begin{corollary} \label{coro2.6} Under assumptions {\rm (A1), (A2), (A4)}, we have \begin{equation}\label{2.3} \Phi(u) \geq \Phi(tu)+\frac{1-t^2}{2}\langle\Phi'(u),u\rangle, \quad \forall u \in X(\Omega),\; t\geq0. \end{equation} \end{corollary} \begin{corollary} \label{coro2.7} Under assumptions {\rm (A1),(A2), (A4)}, we have \begin{equation}\label{2.4} \Phi(u)\geq\Phi(su^{+}+tu^{-}),\quad \forall u \in \mathcal{M},\; s,t\geq0. \end{equation} \end{corollary} \begin{lemma} \label{lem2.8} Assume {\rm (A1)--(A4)} hold; if $u\in X(\Omega)$ with $u^{\pm}\neq0$, then there exists a unique pair $(s_u,t_u)$ of positive numbers such that $s_uu^{+}+t_uu^{-}\in\mathcal{M}$. \end{lemma} \begin{proof} Let \begin{gather}\label{2.5} \begin{aligned} g_1(s,t)&=\langle\Phi'(su^{+}+tu^{-}),su^{+}\rangle\\ &=s^2\|u^{+}\|^2-\int_{\Omega}f(x,su^{+})su^{+}\mathrm{d}x+2B(u)st, \end{aligned}\\ \label{2.6} \begin{aligned} g_2(s,t)&=\langle\Phi'(su^{+}+tu^{-}),tu^{-}\rangle \\ &=t^2\|u^{-}\|^2-\int_{\Omega}f(x,tu^{-})tu^{-}\mathrm{d}x+2B(u)st. \end{aligned} \end{gather} From (A1), (A2) and (A3), a straightforward computation yields that there are $r>0$ small enough and $R>0$ large enough such that \begin{gather*} g_1(r,r)>0,\quad g_2(r,r)>0, \\ g_1(R,R)<0,\quad g_2(R,R)<0. \end{gather*} Notice that for any fixed $s>0$, $g_1(s,t)$ is increasing in $t$ on $[0,+\infty)$, then \begin{gather*} g_1(r,t)\geq g_1(r,r)>0,\quad \forall t\in[r,R], \\ g_1(R,t)\leq g_1(R,R)<0,\quad \forall t\in[r,R]. \end{gather*} Analogously, for $g_2(s,t)$, one has \begin{gather*} g_2(s,r)\geq g_2(r,r)>0,\quad \forall s\in[r,R], \\ g_2(s,R)\leq g_2(R,R)<0,\quad \forall s\in[r,R]. \end{gather*} The above inequalities and the Miranda theorem \cite{MC} imply that there is a pair $(s_u,t_u)\in(r,R)\times(r,R)$ such that $g_1(s_u,t_u)=g_2(s_u,t_u)=0$, and then, $s_uu^{+}+t_uu^{-}\in\mathcal{M}$. Next, we prove the uniqueness. Let $(\hat{s}_1,\hat{t}_1)$ and $(\hat{s}_2,\hat{t}_2)$ such that $\hat{s}_{i}u^{+}+\hat{t}_{i}u^{-}\in\mathcal{M}$, $i=1,2$. We assume that $(\frac{\hat{s}_2}{\hat{s}_1}-1)^2+(\frac{\hat{t}_2}{\hat{t}_1}-1)^2\neq0$, then Lemma \ref{lem2.5} implies \begin{gather*} \Phi(\hat{s}_1u^{+}+\hat{t}_1u^{-})>\Phi(\hat{s}_2u^{+}+\hat{t}_2u^{-}),\\ \Phi(\hat{s}_2u^{+}+\hat{t}_2u^{-})>\Phi(\hat{s}_1u^{+}+\hat{t}_1u^{-}). \end{gather*} This contradiction shows $(\hat{s}_1,\hat{t}_1)=(\hat{s}_2,\hat{t}_2)$, this completes the proof. \end{proof} \begin{corollary} \label{coro2.9} Under assumptions {\rm (A1)--(A4)}, \[ m:=\inf_{u\in\mathcal{M}}\Phi(u) =\inf_{u\in X(\Omega),u^{\pm}\neq0}\max_{s,t\geq0}\Phi(su^{+}+tu^{-}). \] \end{corollary} \begin{lemma} \label{lem2.10} Assume that {\rm (A1)--(A4)} hold. If $u_{0}\in\mathcal{M}$, and $\Phi(u_{0})=m$, then $u_{0}$ is a critical point of $\Phi$. \end{lemma} \begin{proof} Arguing by contradiction, $\Phi(u_{0})=m$ and $\Phi'(u_{0})\neq0$. Therefore, there exist $\delta>0$ and $\rho>0$ such that \[ v\in X(\Omega),\; \|v-u_{0}\|\leq3\delta \Rightarrow\|\Phi'(v)\|\geq\rho. \] Let $D=(\frac{1}{2},\frac{3}{2})\times(\frac{1}{2},\frac{3}{2})$. It follows from Lemma \ref{lem2.5} that \[ \bar{m}:=\max_{(s,t)\in\partial D}\Phi(su_{0}^{+}+tu_{0}^{-})m+2\varepsilon$; \item[(ii)] $\eta(1,\Phi^{m+\varepsilon}\cap B(u_{0},\delta)) \subset\Phi^{m-\varepsilon}$; \item[(iii)] $\Phi(\eta(1,u))\leq\Phi(u)$, for all $u\in X(\Omega)$. \end{itemize} By Corollary \ref{coro2.7}, $\Phi(su_{0}^{+}+tu_{0}^{-})\leq\Phi(u_{0})=m$, for $s,t\geq0$, then from (ii) it follows that \begin{equation}\label{2.7} \Phi(\eta(1,su_{0}^{+}+tu_{0}^{-}))\leq m-\varepsilon,\quad \forall s,t\geq0,\; |s-1|^2+|t-1|^2<\frac{\delta^2}{2\|u_{0}\|_{X}^2}. \end{equation} On the other hand, by (iii) and Lemma \ref{lem2.5}, one has \begin{equation}\label{2.8} \Phi(\eta(1,su_{0}^{+}+tu_{0}^{-}))\leq\Phi(su_{0}^{+}+tu_{0}^{-})<\Phi(u_{0}) =m, \end{equation} for all $s,t\geq0$, $|s-1|^2+|t-1|^2\geq\frac{\delta^2}{2\|u_{0}\|_{X}^2}$. Combining \eqref{2.7} and \eqref{2.8}, we have \[ \max_{(s,t)\in\bar{D}}\Phi(\eta(1,su_{0}^{+}+tu_{0}^{-}))0$ can be achieved to get a critical point of $\Phi$. Let ${u_{n}}$ be a sequence in $\mathcal{M}$ such that \[ \lim_{n\to \infty}\Phi(u_{n})=m. \] First of all, we claim that $\{u_{n}\}$ is bounded in $X(\Omega)$. To this end, suppose by contradiction that $\|u_{n}\|_{X} \to \infty $, and set $v_{n}=\frac{u_{n}}{\|u_{n}\|}$. Since $\|v_{n}\|_{X}=1$, passing to a subsequence, there exists $v\in X(\Omega)$ such that $v_{n}\rightharpoonup v$ in $X(\Omega)$, $v_{n}\to v$ in $L^{p}(\Omega)$, for $2\leq p<2_{\alpha}^{*}$, and $v_{n}(x)\to v(x)$ a.e. on $\Omega$. If $v=0$, then we have $v_{n}\to 0$ in $L^{p}(\Omega)$, for $2\leq p<2_{\alpha}^{*}$. Fix $\tau\in [2,2_{\alpha}^{*})$ and $R=\sqrt{2(m+1)}$. By (A1) and (A2), given $\varepsilon>0$, there exists $C_{\varepsilon}>0$, such that \begin{equation}\label{3.1} |F(x,s)|\leq\varepsilon|s|^2+C_{\varepsilon}|s|^{\tau} +\varepsilon|s|^{2_{\alpha}^{*}},\quad \forall x\in\Omega,s\in\mathbb{R}. \end{equation} By \eqref{3.1}, Corollary \ref{coro2.6} and Lebesgue's dominated convergence theorem, it follows that \begin{align*} m&=\Phi(u_{n})+o(1)\\ &\geq\Phi(Rv_{n})+\Big(\frac{1}{2}-\frac{R^2}{2\|u_{n}\|^2}\Big)\langle \Phi'\langle u_{n}),u_{n}\rangle+o(1)\\ &=\frac{R^2}{2}-\int_{\Omega}F(x,Rv_{n})\mathrm{d}x+o(1) \\ &\geq\frac{R^2}{2}-\int_{\Omega}|F(x,Rv_{n})|\mathrm{d}x+o(1) \\ &\geq\frac{R^2}{2}- \int_{\Omega}\big[\varepsilon|Rv_{n}|^2 +C_{\varepsilon}|Rv_{n}|^{\tau}+\varepsilon|Rv_{n}|^{2_{\alpha}^{*}}\big]\mathrm{d}x +o(1) \\ &=m+1-\big\{\varepsilon\big[R^2\|v_{n}\|_2^2+R^{2_{\alpha}^{*}}\|v_{n} \|^{2_{\alpha}^{*}}_{{2_{\alpha}^{*}}}\big]+ C_{\varepsilon}R^{\tau}\|v_{n}\|_{\tau}^{\tau}\big\}+o(1)\\ &\geq m+1-C_1\varepsilon+o(1), \end{align*} the contradiction is obvious due to the arbitrariness of $\varepsilon$. Thus, $v\neq0$. Denote $A=\{x\in\Omega:v(x)\neq0\}$. Then for $x\in A$, we have $\lim_{n\to \infty}|u_{n}(x)|=\infty$. By (A3), (A4) and Fatou's Lemma \begin{align*} 0&=\lim_{n\to \infty}\frac{\Phi(u_{n})}{\|u_{n}\|^2}\\ &=\lim_{n\to \infty}\Big[\frac{1}{2}-\int_{A}\frac{F(x,u_{n})}{u_{n}^2} v_{n}^2\mathrm{d}x\Big]\\ &\leq\frac{1}{2}-\liminf_{n\to \infty}\int_{A}\frac{F(x,u_{n})}{u_{n}^2} v_{n}^2\mathrm{d}x\\ &\leq\frac{1}{2}-\int_{A}\liminf_{n\to \infty} \frac{F(x,u_{n})}{u_{n}^2} v_{n}^2\mathrm{d}x=-\infty. \end{align*} The contradiction shows that $\{u_{n}\}$ is bounded in $X(\Omega)$. Passing to a subsequence, there exists $u\in X(\Omega)$ such that $u_{n}\rightharpoonup u~\mathrm{in}~X(\Omega)$, $u_{n}\to u~\mathrm{in}~L^{p}(\Omega)$, for $2\leq p<2_{\alpha}^{*}$, and $u_{n}(x)\to u(x)$ a.e. on $\Omega$. Next, we show that $m>0$ is attained. From Lemma \ref{lem2.4}, it follows that $u^{\pm}\neq0$. Then by Lemma \ref{lem2.8}, there are $s,t>0$ such that $su^{+}+tu^{-}\in\mathcal{M}$. By Lemmas \ref{lem2.3} and \ref{lem2.5}, we have \begin{align*} m&\leq\Phi(su^{+}+tu^{-})\\&\leq\Phi(u) -\frac{1-s^2}{2}\langle\Phi'(u),u^{+}\rangle -\frac{1-t^2}{2}\langle\Phi'(u),u^{-}\rangle\\ &=\Phi(u)-\frac{1}{2}\langle\Phi'(u),u\rangle +\frac{s^2}{2}\langle\Phi'(u),u^{+}\rangle +\frac{t^2}{2}\langle\Phi'(u),u^{-}\rangle\\ &\leq\lim_{n\to \infty}\int_{\Omega} \big[\frac{1}{2}f(x,u_{n})-F(x,u_{n})\big]\mathrm{d}x \\ &\quad +\liminf_{n\to \infty}\big\{\frac{s^2}{2}\langle\Phi'(u_{n}),u_{n}^{+}\rangle +\frac{t^2}{2}\langle\Phi'(u_{n}),u_{n}^{-}\rangle\big\}\\ &=\lim_{n\to \infty}\big[\Phi(u_{n})-\frac{1}{2}\langle\Phi'(u_{n}),u_{n}\rangle\big] =m, \end{align*} which implies $\Phi(su^{+}+tu^{-})=m$. From Lemma \ref{lem2.10}, $\Phi'(su^{+}+tu^{-})=0$, and then $su^{+}+tu^{-}$ is a sign-changing solution of \eqref{1.1}. \end{proof} \subsection*{Acknowledgments} This work is partially supported by grants from the NNSF (Nos: 11571370, 11471137, 11471278). \begin{thebibliography}{00} \bibitem{GPE} G. Autuori, P. Pucci; \emph{Elliptic problems involving the fractional Laplacian in $\mathbb{R}^{N}$}, J. Differ. Equ. 255 (2013), 2340-2362. \bibitem{BC} T. Bartsch, K. C. Chang, Z. Q. Wang; \emph{On the Morse indices of sign-changing solutions for nonlinear elliptic problems}, Math. Z. 233 (2000), 655-677. \bibitem{BW} T. Bartsch, Z. Q. Wang; \emph{On the existence of sign-changing solutions for semilinear Dirichlet problems}, Topl. Methods Nonlinear Anal. 7 (1996), 115-131. \bibitem{XY1} X. Cabre, Y. Sire; \emph{Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates}, Ann. Inst. H. Poincar¡äe Anal. Non Lin¡äeaire 31 (2014), 23-53. \bibitem{XY2} X. Cabre, Y. Sire; \emph{Nonlinear equations for fractional Laplacians, II: existence, uniqueness, and qualitative properties of solutions}, Trans. Amer. Math. Soc. 367 (2015), 911-941. \bibitem{LC1} L. Caffarelli; \emph{Surfaces minimizing nonlocal energies}, AttiAccad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 20 (2009), 281-299. \bibitem{LC3} L. Caffarelli, L. Silvestre; \emph{An extension problem related to the fractional Laplacian}, Commun. Partial Differ. Equations 32 (2007), 1245-1260. \bibitem{LC2} L. Caffarelli, J. L. Vazquez; \emph{Nonlinear porous medium flow with fractional potential pressure}, Arch. Ration. Mech. Anal. 202 (2011), 537-565. \bibitem{CM} M. Cheng; \emph{Bound state for the fractional Schr\"{o}dinger equation with unbounded potentials}, J. Math. Phys. 53 (2012), 043507. \bibitem{CXJ} X. J. Chang; \emph{Ground state solutions of asymptotically linear fractional Schr\"{o}dinger equations}, J. Math. Phys. 54 (2013), 061504. \bibitem{CX} X. Chang, Z. Q. Wang; \emph{Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity}, Nonlinearity 26 (2013), 479-494. \bibitem{CW} X. J. Chang, Z. Q. Wang; \emph{Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian}, J. Differential Equations 256 (2014), 2965-2992. \bibitem{CMT} M. Conti, L. Merizzi, S. Terracini; \emph{Remarks on variational method and lower-upper solutions}, NoDEA Nonlinear Differ. Equ. Appl. 6 (1999), 371-393. \bibitem{SGE} S. Dipierro, G. Palatucci, E. Valdinoci; \emph{Existence and symmetry results for a Schr\"{o}dinger type problem involving the fractional Laplacian}, Matematiche (Catania) 68 (2013), 201-216. \bibitem{PAJ} P. Felmer, A. Quaas, J. Tan; \emph{Positive solutions of nonlinear Schr\"{o}dinger equation with the fractional Laplacian}, Proc. R. Soc. Edinburgh Sect. A 142 (2012), 1237-1262. \bibitem{GH} B. Guo, D. Huang; \emph{Existence and stability of standing waves for nonlinear fractional Schr\"{o}dinger equations}, J. Math. Phys. 53 (2013) 083702. \bibitem{K} S. Khoutir, H. Chen; \emph{Existence of infinitely many high energy solutions for a fractional Schr\"{o}dinger equation in $\mathbb{R}^{N}$}, Applied Mathematics Letters 61 (2016), 156-162. \bibitem{NL} N. Laskin; \emph{Fractional Schr\"{o}dinger equation}, Phys. Rev. 66 (2002), 56-108. \bibitem{LT} X. Y. Lin, X. H. Tang; \emph{Nehari-Type ground state positive solutions for superlinear asymptotically periodic for Schr\"{o}dinger equations}, Abstr Appl Anal. (2014) ID 607078, 7pp. \bibitem{LT1} X. Y. Lin, X. H. Tang; \emph{An asymptotically periodic and asymptotically linear Schr\"{o}dinger equation with indefinite linear part}, Comput. Math. Appl. 70 (2015), 726-736. \bibitem{LLS} Z. L. Liu, J. X. Sun; \emph{Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations}, J. Differential Equations 172 (2001), 257-299. \bibitem{LZ} Z. L. Liu, Z. Q. Wang; \emph{Sign-changing solutions of nonlinear elliptic equations}, Front. Math. China 3 (2008)1-18. \bibitem{MC} C. Miranda; \emph{Un' osservazione su un teorema di Brouwer}. Bol. Un. Mat. Ital. 3 (1940) 5-7. \bibitem{EGE} E. D. Nezza, G. Palatucci, E. Valdinoci; \emph{Hitchhiker's guide to the fractioal Sobolev spaces}, Bull. Sci. Math. 136 (2012) 521-573. \bibitem{SS} S. Secchi; \emph{Ground state solutions for nonlinear fractional Schr\"{o}dinger equations in $\mathbb{R}^{N}$}, J. Math. Phys. 54 (2013) 031501. \bibitem{SV} R. Servadei, E. Valdinoci; \emph{Mountain pass solutions for non-local elliptic operators}, J. Math. Anal. Appl. 389 (2012), 887-898. \bibitem{S} H. X. Shi, H. B. Chen; \emph{Multiple solutions for fractional Schr\"{o}dinger equations}, Elect. J. Differential Equations 25 (2015), 1-11. \bibitem{SW} W. Shuai; \emph{Sign-changing solitions for a class of Kirchhoff-type problems in bounded domains}, J. Differential Equations 259 (2015) 1256-1274. \bibitem{LS} L. Silvestre; \emph{Regularity of the obstacle problem for a fractional power of the Laplace operator}, Comm. Pure Appl. Math. 60 (2007), 67-112. \bibitem{TWY} J. Tan, Y. Wang, J. Yang; \emph{Nonlinear fractional field equations}, Nonlinear Anal.75 (2012) 2098-2110. \bibitem{TJ} J. Tan; \emph{The Brezis-Nirenberg type problem involving the square root of the Laplacian}, Calc. Var. Partial Differ. Equat. 42 (2011), 21-41. \bibitem{T1} X. H. Tang; \emph{New conditions on nonlinearity for a periodic Schr\"{o}dinger equation having zero as spectrum}, J. Math. Anal. Appl. 413 (2014), 392-410. \bibitem{T2} X. H. Tang; \emph{Non-nehari-manifold method for asymptotically linear Schr\"{o}dinger equation}, J. Aust. Math. Soc. 98 (2015), 104-116. \bibitem{T3} X. H. Tang; \emph{Non-nehari-manifold method for superlinear Schr\"{o}dinger equation}, Taiwanese J. Math. 18 (2014), 1957-1979. \bibitem{T4} X. H. Tang; \emph{Non-nehari manifold method for asymptotically periodic Schr\"{o}dinger equations}, Sci. China Math. 58 (2015) 715-728. \bibitem{T5} X. H. Tang; \emph{New super-quadratic conditions on ground state solutions for superlinear Schr\"{o}dinger equation}, Adv. Nonlinear Stud. 14 (2014), 349-361. \bibitem{TKM} K. M. Teng; \emph{Multiple solutions for a class of fractional Schr¡§odinger equations in $\mathbb{R}^{N}$}, Nonlinear Anal. Real World Appl. 21 (2015), 76-86. \bibitem{TWW} K. M. Teng, K. Wang, R. M. Wang; \emph{A sign-changing solution for nonlinear problems involving the fractional Laplacian}, Elect. J. Differential Equations, 2015 (2015), 1-12. \bibitem{W} M. Willem; \emph{Minimax Theorems}. Birkh\"{a}user, Basel (1996). \bibitem{YR} R. Yang; \emph{Optimal regularity and nondegeneracy of a free boundary problem related to the fractioal Laplacian}, Arch. Ration. Mech. Anal. 208 (2013,) 693-723. \bibitem{ZTZ} W. Zhang, X. H. Tang, J. Zhang; \emph{Infinitely many radial and non-radial solutions for a fractional Schr\"{o}dinger equation}, Comput. Math. Appl. 71 (2016), 737-747. \bibitem{ZTZ1} J. Zhang, X. H. Tang, W. Zhang; \emph{Infinitely many solutions of quasilinear Schr\"{o}dinger equation with sign-changing potential}, J. Math. Anal. Appl. 420 (2014), 1762-1775. \bibitem{ZXZ} H. Zhang, J. X. Xu, F. B. Zhang; \emph{Existence and multiplicity of solutions for superlinear fractional Schr\"{o}dinger equations in $\mathbb{R}^{N}$}, J. Math. Phys. 56 (2015), 091502. \end{thebibliography} \end{document}