\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 05, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/05\hfil Fractional Schr\"odinger equations] {Fractional Schr\"odinger equations with new conditions} \author[A. Benhassine \hfil EJDE-2018/05\hfilneg] {Abderrazek Benhassine} \address{Abderrazek Benhassine \newline Department of Mathematics, Higher Institut for Informatics and Mathematics, 5000, Monastir, Tunisia} \email{ab.hassine@yahoo.com} \dedicatory{Communicated by Raffaella Servadei} \thanks{Submitted July 28, 2017. Published January 4, 2018.} \subjclass[2010]{35B38, 35G99} \keywords{Fractional Schr\"odinger equations; critical point theory; \hfill\break\indent symmetric mountain pass theorem} \begin{abstract} In this article, we study the nonlinear fractional Schr\"odinger equation \begin{gather*} (-\Delta)^{\alpha}u+ V(x)u= f(x,u)\\ u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}), \end{gather*} where $(-\Delta)^{\alpha}(\alpha \in (0, 1))$ stands for the fractional Laplacian of order $\alpha$, $x\in \mathbb{R}^{n}$, $V\in C(\mathbb{R}^{n},\mathbb{R})$ may change sign and $f$ is only locally defined near the origin with respect to $u$. Under some new assumptions on $V$ and $f$, we show that the above system has infinitely many solutions near the origin. Some examples are also given to illustrate our main theoretical result. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction and statement of main results} This article concerns the existence of infinitely many solutions for the fractional Schr\"odinger equation \begin{equation}\label{eq1.1} \begin{gathered} (-\Delta)^{\alpha}u+ V(x)u= f(x,u), \\ u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}), \end{gathered} \end{equation} where $n\geq2$, $\alpha \in (0, 1)$, $x\in \mathbb{R}^{n}$, $V \in C(\mathbb{R}^{n},\mathbb{R})$ satisfying some new conditions, and $f$ is only locally defined near the origin with respect to $u$. Problem \eqref{eq1.1} is related to the existence of standing wave solutions for fractional Schr\"odinger equations of the form \begin{equation}\label{eq1.2} i\frac{\partial \psi}{\partial t} = (-\Delta)^{\alpha}\psi +( V(x)+\omega)\psi- f(x,\psi), \end{equation} where $i$ is the imaginary unit, $\alpha \in (0, 1)$, $\omega$ is a constant, $(-\Delta)^{\alpha}$ is the fractional Laplacian operator of order $\alpha$ and $\psi : \mathbb{R}^{3}\times [0,+\infty) \to \mathbb{C}$. We are interested in looking for a standing wave, namely, waves of the form $$ \psi(x,t)=e^{i \omega t}u(x), $$ where $u$ is a real-valued function, and $f$ is assumed to satisfy $f(x, e^{-i\omega t}u)=e^{-i\omega t} f(x,u)$. Clearly, $\psi(x, t)$ solves \eqref{eq1.2} if and only if $u(x)$ solves \eqref{eq1.1}. The fractional Schr\"odinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin \cite{L, L1} as a result of extending the Feynman path integral, from the Brownian-like to L\'evy-like quantumn mechanical paths. Equations involving the fractional Laplacian have attracted much attention in recent years, appear in several areas such as optimization, finance, phase transitions, stratified material, crystal dislocation, flame propagation, conservation laws, ultra-relativistic limits of quantum, material science, and water waves, see e.g. \cite{B0, BRS, C0, DPV} for an introduction to these topics and their applications. When $\alpha=1$, \eqref{eq1.1} becomes the classical Schr\"odinger equation \begin{equation}\label{eq1.3} \begin{gathered} -\Delta u+V(x)u= f(x,u)\\ u\in H^{1}(\mathbb{R}^{n},\mathbb{R}). \end{gathered} \end{equation} There has been a a lot of studies on existence and multiplicity of solutions of problem \eqref{eq1.3} under various hypotheses on the potential $V(x)$ and the nonlinearity $f(x, u)$, see \cite{BT, FQT, R, R1} and the references therein. The body of literature for \eqref{eq1.3} is huge and we do not even try to collect here a detailed bibliography. Nonlinear equation \eqref{eq1.1} involves the fractional Laplacian $ (-\Delta)^{\alpha}$, $0 < \alpha <1$, which is a nonlocal operator. A common approach to deal with this problem was proposed by Caffarelli and Silvestre in \cite{CS}, see also \cite{S0}, allowing to transform problem \eqref{eq1.1} into a local problem via the Dirichlet-Neumann map. That is, for $u \in H^{\alpha}(\mathbb{R}^{n})$ one considers the problem \begin{gather*} -\operatorname{div}(y^{1-2\alpha}\nabla v)=0\quad \text{in } \mathbb{R}^{n+1}_{+}\\ v(x,0)=u,\quad \text{on } \mathbb{R}^{n} \end{gather*} from where the fractional Laplacian is obtained as $$ (-\Delta)^{\alpha}u(x)=-b_{\alpha}\lim_{y\to 0^{+}}y^{1-2\alpha}v_{y} $$ where $b_{\alpha}$ is a suitable constant. With the aid of the extended techniques \cite{CS}, some existence and nonexistence results for Dirichlet problem involving the fractional Laplacian on bounded domain are obtained, see e.g. \cite{CT, T01} and the references therein. Using the equivalence definition of fractional operator $(-\Delta)^{\alpha}$ (see Section 2), Servadei and Valdinoci \cite{SV, SV1} also introduced a variational principle and studied the existence and multiplicity of solutions for non-local equations of elliptic type. There have been many results appeared in the literature for problem \eqref{eq1.1}. For example, Cheng \cite{C} studied problem \eqref{eq1.1} when $f(x,u)=|u|^{p-1}u$ with $1 < p < \frac{4\alpha}{n}+1$, and found the ground states under a stronger assumption on the potential $V$, i.e., $\lim_{|x|\to \infty}V(x)=\infty$. Dipierro et al.\ \cite{DiPV} studied problem \eqref{eq1.1} when the potential $V(x) = 1$ and $f(x,u)=|u|^{p-1}u$, with $1 < p < \frac{2n}{n-2\alpha}$; in this case, they established the existence of positive and spherically symmetric solution. Felmer et al.\ \cite{FQT} studied a similar class of equations, in which $V(x) = 1$, and the nonlinearity satisfies suitable assumptions, using variational methods, classical positive solutions are found. Secchi \cite{S00} proved some existence results for fractional Schr\"odinger equations, under the assumption that the nonlinearity is either of perturbative type or satisfies the Ambrosetti-Rabinowitz condition. Recently, Teng \cite{T01} obtained infinitely many small energy solutions of \eqref{eq1.1} by variant of the fountain theorem in \cite{Z}. More precisely, they use the following assumptions: \begin{itemize} \item[(A1)] $V\in C(\mathbb{R}^n,\mathbb{R})$ and $\inf_{\mathbb{R}^n} V >0$. \item[(A2)] For any $M>0$ there exists $d_0>0$ such that $$ \lim_{|y|\to \infty} \operatorname{meas} (\{x\in \mathbb{R}^{n}: |x-y|\leq d_0, V(x)\leq M\})=0, $$ where meas denotes the Lebesgue measure in $\mathbb{R}^{n}$. \item[(A3)] $f \in C(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R}), f(x,u)u\geq 0$ for all $(x,u)\in \mathbb{R}^{n}\times \mathbb{R}$, and there exists a constant $\nu \in (1,2)$ such that $$ |f(x,u)|\leq a(x)(1+|u|^{\nu-1})\, \forall (x,u)\in \mathbb{R}^{n}\times \mathbb{R} $$ with a positive function $a(x) \in L^{\frac{2}{2-\nu}}(\mathbb{R}^{n})$. \item[(A4)] There exists $\sigma \in [1,\nu)$ such that $\liminf_{|u|\to \infty}\frac{F(x,u)}{|u|^{\sigma}} \geq d>0$ uniformly for $x\in \mathbb{R}^{n}$, where where $F(x,u)= \int^{u}_{0}f(x,s)ds$. \item[(A5)] $f(x,-u)=-f(x,u)$ for all $(x,u)\in \mathbb{R}^{n}\times \mathbb{R}$. \end{itemize} Very recently, Torres \cite{T} studied problem \eqref{eq1.1} and proved the existence of at least one solutions of equation \eqref{eq1.1} under the assumptions: \begin{itemize} \item[(A6)] $V(x)=\lambda v(x)$ where $\lambda>0$ is a parameter and $v\in C(\mathbb{R}^{n}), v(x)\geq 0$ on $\mathbb{R}^{n}$; \item[(A7)] there exists a constant $b>0$ such that the set $\{v0$ such that $V(x)+a_{0}\geq 1$, and $\int_{\mathbb{R}^{n}}\frac{1}{V(x)+a_{0}}dx < \infty$. \item[(A10)] $ F \in C^{1}(\mathbb{R}^{n}\times (-\rho,\rho))$ is even, and there exists a constant $a_1>0$ such that $$ |f(x,u)|\leq a_1, \quad \forall(x,u)\in \mathbb{R}^{n}\times (-\rho,\rho), $$ where $\rho>0$. \item[(A11)] There exist $x_{0}\in \mathbb{R}^{n}$, two sequences of positives numbers $\varepsilon_n\to0$, $M_n\to \infty$ as $n\to \infty$ and constants $a_2, \varepsilon, \delta>0$ such that \begin{gather*} F(x,u)\geq \varepsilon^{2}_n M_n,\quad \text{for } |x-x_{0}|\leq\delta\ \text{and}\ |u|= \varepsilon_n \\ F(x,u)\geq -a_2u^{2},\quad \text{for } |x-x_{0}|\leq\delta \text{ and } |u|\leq \varepsilon. \end{gather*} \end{itemize} Now we give our main results. \begin{theorem} \label{th1.1} Assume that {\rm (A9)--(A11)} are satisfied. Then, equation \eqref{eq1.1} possesses a sequence of solutions $(u_k)$ such that $$ \frac{1}{2}\Big(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}} \frac{|u_k(x)-u_k(z)|^{2}}{|x-z|^{n+2\alpha}} \,dz\,dx +V(x)u^{2}_k\Big)dx -\int_{\mathbb{R}^{n}}F(x,u_k)dx\to 0^{-} $$ as $ k\to \infty$. \end{theorem} \begin{corollary}\label{cr1.1} Assume that {\rm (A9), (A10)} are satisfied and \begin{itemize} \item[(A11')] there exist $ x_{0}\in \mathbb{R}$ and a constant $\delta>0$, such that \begin{gather*} \liminf_{|u|\to0}\inf_{|x-x_{0}|\leq\delta}\frac{F(x,u)}{|u|^{2}}> -\infty, \\ \limsup_{|u|\to0}\inf_{|x-x_{0}|\leq\delta}\frac{F(x,u)}{|u|^{2}}=+\infty. \end{gather*} \end{itemize} Then, equation \eqref{eq1.1} possesses a sequence of solutions $(u_k)$ such that $$ \frac{1}{2}\Big(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}} \frac{|u_k(x)-u_k(z)|^{2}}{|x-z|^{n+2\alpha}} \,dz\,dx +V(x)u^{2}_k\Big)dx -\int_{\mathbb{R}^{n}}F(x,u_k)dx\to 0^{-} $$ as $k\to \infty$. \end{corollary} The remainder part of this article is organized as follows. Some preliminary results are presented in Section 2. In Section 3, we give the proofs of our main results. \section{Variational setting and preliminaries} In this section, we recall some preliminary results which will be useful in this article. First, we will give some facts of the fractional order Sobolev spaces. For any $0<\alpha<1$, the fractional Sobolev space $H^{\alpha}(\mathbb{R}^{n})$ is defined by $$ H^{\alpha}(\mathbb{R}^{n})=\Big\{u\in L^{2}(\mathbb{R}^{n}): \frac{|u(x)-u(z)|}{|x-z|^{\frac{n+2\alpha}{2}}} \in L^{2}(\mathbb{R}^{n}\times\mathbb{R}^{n})\Big\}, $$ endowed with the natural norm $$ \|u\|^{2}_{\alpha}=\int_{\mathbb{R}^{n}}|u(x)|^{2}dx + \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}} \frac{|u(x)-u(z)|^{2}}{|x-z|^{n+2\alpha}} \,dz\,dx. $$ For the reader's convenience, we review the main embedding result for this class of fractional Sobolev spaces. \begin{lemma}[\cite{DPV}] \label{lem2.1} Let $0<\alpha<1$ such that $2\alphaR}|u_n-u|dx\\ &\leq 2 R \Big(\int_{|x|\leq R}|u_n-u|^{2}dx\Big)^{1/2}+ \int_{|x|> R} |( V(x))^{-1/2}( V(x))^{1/2}(u_n-u)|dx\\ &\leq 2R \Big(\int_{|x|\leq R}|u_n-u|^{2}dx\Big)^{1/2} + \int_{|u|>R} (V(x))^{-\frac{1}{2}}|(V(x))^{1/2}(u_n-u)|dx\\ &\leq 2R \Big(\int_{|x|\leq R}|u_n-u|^{2}dx\Big)^{1/2} \\ &\quad + \Big(\int_{|x|>R} ( V(x))^{-1}dx\Big)^{1/2} \Big(\int_{|x|>R} V(x)(u_n-u)^{2}dx\Big)^{1/2}\\ &\leq 2R \Big(\int_{|x|\leq R}|u_n-u|^{2}dx\Big)^{1/2} + \Big(\int_{|x|>R} ( V(x))^{-1}dx\Big)^{1/2}\|u_n-u\|_{V}, \end{aligned} \end{equation} where $R>0$. Since the embedding is compact on bounded domain then, by (A9) and \eqref{eq2.4}, we have $u_n\to u$ in $L^{1}$. \end{proof} \section{Proofs of main resutls} The aim of this section is to establish the proofs of Theorem \ref{th1.1} and Corollary \ref{cr1.1}. For this purpose, we need to modify $F(x,u)$ for $u$ outside a neighborhood of the origin to get a globally defined $\widetilde{F}(x,u)$ as follows: Choose a constant $t_{0} \in (0,\frac{\rho}{2})$ and define a cut-off function $\chi \in C^{1}(\mathbb{R}^{+},\mathbb{R}^{+})$ satisfying \begin{equation}\label{eq3.1} \begin{gathered} \chi(t)=\begin{cases} 1 & \text{if } 0\leq t\leq t_{0}\\ 0 & \text{if } t\geq 2t_{0} \end{cases}\\ -\frac{2}{t_{0}}\leq \chi'(t) <0 \quad \text{for } t_{0}< t< 2t_{0}. \end{gathered} \end{equation} Let $\widetilde{F}(x,u)= \chi(|u|) F(x,u)$, for all$(x,u)\in \mathbb{R}^{n}\times \mathbb{R}$. By \eqref{eq3.1} and (A10) we have, for all $(x,u)\in \mathbb{R}^{n}\times \mathbb{R}$, \begin{equation}\label{eq3.2} |\widetilde{F}(x,u)| \leq c_1|u| , \quad | \widetilde{f}(x,u)| \leq c_2. \end{equation} Now we consider the modified fractional Schr\"odinger equation \begin{equation}\label{eq3.3} \begin{gathered} (-\Delta)^{\alpha}u+ V(x)u= \widetilde{f}(x,u), \\ u\in H^{\alpha}(\mathbb{R}^{n},\mathbb{R}), \end{gathered} \end{equation} Define the functional $I: H^{\alpha}_{V} \to \mathbb{R}$ associated with \eqref{eq3.3} by \begin{equation} \label{eq3.4} \begin{aligned} I(x) &=\frac{1}{2}\Big(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}} \frac{|u(x)-u(z)|^{2}}{|x-z|^{n+2\alpha}} \,dz\,dx + \int_{\mathbb{R}^{n}}V(x)|u(x)|^{2}dx \Big) \\ &\quad - \int_{\mathbb{R}^{n}}\widetilde{F}(x,u(x))dx \\ &= \frac{1}{2}\|u\|^{2}_{V} - \int_{\mathbb{R}^{n}} \widetilde{F}(x,u(x))dx. \end{aligned} \end{equation} Then, by (A9), (A10) and \eqref{eq3.2}, we see that $I$ is a continuously Fr\'echet-differentiable functional defined on $H^{\alpha}_{V}$; i.e., $I\in C^{1}(H^{\alpha}_{V}, \mathbb{R})$. Moreover, we have \begin{equation}\label{eq3.5} \begin{aligned} I'(u)v &= \int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}} \frac{[u(x)-u(z)][v(x)-v(z)]}{|x-z|^{n+2\alpha}} \,dz\,dx + \int_{\mathbb{R}^{n}}V(x)u(x)v(x)dx \\ &\quad - \int_{\mathbb{R}^{n}}\widetilde{f}(x,u(x))v(x) dx, \end{aligned} \end{equation} for all $u,v \in H^{\alpha}_{V}$. According to \cite{T}, we know that in order to find solutions of \eqref{eq3.3}, it suffices to obtain the critical points of $I$. For this purpose we recall the following definitions and results (see \cite{KJ, R}). \begin{definition} \rm Let $E$ be a real Banach space and $\phi \in C^1(E,\mathbb{R})$. $\bullet$ $\phi$ is said to satisfy the (PS) condition if any sequence $(x_k)\subset E$ for which $(\phi(x_k))$ is bounded and $\phi'(x_k)\to 0$ as $k \to +\infty$, possesses a convergent subsequence in $E$. $\bullet$ Set $\Sigma =\{A\subset E \backslash\{0\} : A \text{ is closed and symmetric with respect to the origin}\}$. For $A \in \Sigma$, we say genus of $A$ is $n$ (denoted by $\kappa (A)=n$), if there is an odd map $\varphi \in C(A,\mathbb{R}^{n}\backslash\{0\})$, and $n$ is the smallest integer with this property. \end{definition} \begin{lemma}[{\cite[Theorem 1]{KJ}}]\label{lem3.2} Let $\phi$ be an even $C^{1}$ functional on $E$ with $\phi(0)=0$. Suppose that $\phi$ satisfies the $(PS)$ condition and \begin{itemize} \item[(1)] $\phi$ is bounded from below. \item[(2)] For each $k\in \mathbb{N}$, there exists an $A_k\in \Sigma_k$ such that $\sup_{x\in A_k} \phi(x)<0$, where $\Sigma_k=\{A\in \Sigma: \kappa(A)\geq k\}$. \end{itemize} Then either $(i)$ or $(ii)$ below holds. \begin{itemize} \item[(i)] There exists a sequence $(x_k)$ of critical point such that $\phi(x_k)<0$ and $\lim_{k\to\infty}x_k=0$. \item[(ii)] There exists two sequences of critical points $(x_k)$ and $(y_k)$ such that $\phi(x_k)=0, x_k\neq 0,\ \lim_{k\to\infty}x_k=0$, $\phi(y_k)<0, \lim_{k\to\infty}\phi(y_k)=0$, and $(y_k)$ converges to a non-zero limit. \end{itemize} \end{lemma} \begin{lemma}\label{lem3.3} If {\rm (A9), (A10)} are satisfied, then $I$ is bounded from below and satisfies the $(PS)$ condition. \end{lemma} \begin{proof} By (A10), \eqref{eq2.3}, \eqref{eq3.2} and the H\"older inequality, we have, for all $u\in H^{\alpha}_{V}$, \begin{equation}\label{eq3.6} \begin{aligned} I(u) &\geq \frac{1}{2}\|u\|^{2}_{V}- c_3 \int_{\mathbb{R}^{n}} |u|dx\cr &\geq \frac{1}{2}\|u\|^{2}_{V}- c_3 \Big(\int_{\mathbb{R}^{n}}(V(x))^{-1}dx\Big)^{1/2} \|u\|_{V}. \end{aligned} \end{equation} Then it follows that $I$ is bounded from below. Moreover, if we take $(u_n)\subset H^{\alpha}_{V}$ be a $(PS)$-sequence, then by \eqref{eq3.2} and \eqref{eq3.4}, we have $$ c_{4}\geq \frac{1}{2}\|u_n\|^{2}_{V} -c_{5}\Big( \int_{\mathbb{R}^{n}}(V(x))^{-1}dx\Big)^{1/2} \|u\|_{V} $$ This implies that $(u_n)$ is bounded in $H^{\alpha}_{V}$. Thus there exists a subsequence $(u_{n_k})$ such that $u_{n_k}\rightharpoonup u_{0}$ as $k\to \infty$ for some $u_{0}\in H^{\alpha}_{V}$. By Lemma \ref{lem2.5}, it holds that $$ u_{n_k}\to u_{0}\quad \text{in } L^{1}\ \text{as}\ k\to \infty. $$ This together with \eqref{eq3.2} yields \begin{equation}\label{eq3.7} \big|\int_{\mathbb{R}^{n}}(\widetilde{f}(x,u_{n_k}) -\widetilde{f}(x,u_{0}))(u_{n_k}-u_{0})dx\big| \leq c_{6}\int_{\mathbb{R}^{n}}|u_{n_k}-u_{0}|dx\to 0 \end{equation} as $k\to \infty$. Noting that $(u_n)$ is a bounded $(PS)$-sequence, we have \begin{equation}\label{eq3.8} (I'(u_{n_k})-I'(u_{0}))(u_{n_k}-u_{0})\to0\ \text{as}\ k\to\infty. \end{equation} Combining \eqref{eq3.5}, \eqref{eq3.7} and \eqref{eq3.8}, we obtain \begin{align*} \|u_{n_k}-u_{0}\|^{2}_{V} &= (I'(u_{n_k})-I'(u_{0}))(u_{n_k}-u_{0}) \\ &\quad +\int_{\mathbb{R}^{n}}(\widetilde{f}(x,u_{n_k}) -\widetilde{f}(x,u_{0})).(u_{n_k}-u_{0})dx\to 0. \end{align*} \end{proof} \begin{proof}[Proof of Theorem \ref{th1.1}] For simplicity, we assume that $x_{0}=0$ in (A11). For $r>0$, let $$ D(r):=\{(x_1,x_2,x_3,\dots,x_n): 0\leq x_i\leq r, i=1,2,3,\dots ,n\}. $$ Fix $r>0$ small enough such that $D(r)\subset B(0,\delta)$, where $\delta$ is the constant given in (A11). For arbitrary $k\in \mathbb{N}$, we construct an $A_k \in \sum_k$ satisfying $\sup_{u\in A_{K}}I(u)<0$. Indeed, we follow the idea of dealing with elliptic problems in Kajikiya \cite{KJ}. Let $m\in \mathbb{N}$ be the smallest integer such that $m^{n}\geq k$. We divide $D(r)$ equally into $m^{n}$ small cubes by planes parallel to each face of $D(r)$ and denote them by $D_i$ with $1\leq i\leq m^{n}$. We consider a cube $E_i\subset D_i\ (i=1,2,\dots ,k)$ such that $E_i$ has the same center as that of $D_i$, the faces of $E_i$ and $D_i$ are parallel and the edge of $E_i$ has length $\frac{a}{2}$. Define $\xi \in C^{\infty}_{0}(\mathbb{R}, [0,1])$ such that $\xi(t)=1$ for $ t\in [\frac{a}{4} , \frac{3a}{4}]$, $\xi(t)=0$ for $t \in (-\infty,0]\cup [a,+\infty)$. Define $$ \zeta(x)=\xi(x_1)\xi(x_2)\xi(x_3)\dots \xi(x_n), (x_1,x_2,x_3,\dots ,x_n)\in \mathbb{R}^{n}. $$ Then $\operatorname{supp} \zeta\subset [0,a]^{n}$. Now for each $1\leq i\leq k$, we can choose a suitable $y_i\in \mathbb{R}^{n} $ and define $$ \zeta_i(x)=\zeta(x-y_i), \quad \text{for all } x \in \mathbb{R}^{n}; $$ such that \begin{equation}\label{eq3.9} \operatorname{supp} \zeta_i\subset D_i, \quad \operatorname{supp} \zeta_i\cap \operatorname{supp} \zeta_{j}=\emptyset\quad (i\neq j), \end{equation} and \begin{equation*} \zeta_i(t)=1, \quad \forall x \in E_i,\; 0\leq\zeta_i(x)\leq 1, \; \forall x \in \mathbb{R}^{n} \end{equation*} Set \begin{equation}\label{eq3.10} \begin{gathered} \Theta_k\equiv \big\{(l_1,l_2, \dots ,l_k) \in \mathbb{R}^{k}; \max_{1\leq i\leq k}|l_i|=1\big\}, \\ S_k\equiv \Big\{\sum^{k}_{i=1}l_i\zeta_i; (l_1,l_2, \dots ,l_k) \in \Theta_k \Big\}. \end{gathered} \end{equation} Then $\Theta_k$ is homeomorphic to the unit sphere in $\mathbb{R}^{k}$ by an odd mapping. Thus $\kappa(\Theta_k)=k$. If we define the following odd and homeomorphic mapping: $\psi: \Theta_k\to S_k$ by $$ \psi(l_1,l_2, \dots ,l_k)= \sum^{k}_{i=1}l_i\zeta_i,\quad \forall (l_1,l_2, \dots ,l_k)\in \Theta_k, $$ Then $\kappa(S_k)=\kappa(\Theta_k)=k$. Moreover, it is evident that $S_k$ is compact and hence there is a constant $\lambda_k>0$ such that \begin{equation}\label{eq3.11} \|u\|_{V}\leq \lambda_k,\quad \forall u \in S_k. \end{equation} For any $s \in (0,\varepsilon)$, $u= \sum^{k}_{i=1}l_i\zeta_i \in S_k$ and by \eqref{eq3.2} and \eqref{eq3.4}, we have \begin{equation}\label{eq3.12} \begin{aligned} I(su) &\leq \frac{s}{2}\|x\|_{V}^{2}-\int_{\mathbb{R}^{n}} F\Big(x, s\sum^{k}_{i=1}l_i\zeta_i\Big)dx\\ &\leq \frac{s^{2}\lambda^{2}_k}{2} -\sum^{k}_{i=1}\int_{D_i} F(x, sl_i\zeta_i)dx. \end{aligned} \end{equation} By \eqref{eq3.10}, there exists an integer $i_{0}\in [1,k]$ such that $|l_{i_{0}}|=1$. Then it follows that \begin{equation}\label{eq3.13} \begin{aligned} \sum^{k}_{i=1}\int_{D_i} F(x, s l_i\zeta_i)dx &= \int_{E_{i_{0}}} F(x, sl_{i_{0}}\zeta_{i_{0}})dx +\int_{D_{i_{0}}\backslash{E_{i_{0}}}} F(x, sl_{i_{0}}\zeta_{i_{0}})dx \\ &\quad + \sum_{i\neq i_{0}}\int_{D_i} F(x, sl_i\zeta_i)dx. \end{aligned} \end{equation} Noting that $|l_{i_{0}}|=1$, $\zeta_{i_{0}}\equiv1$ on $E_{i_{0}}$, and $F(x,u)$ is even in $u$, we get \begin{equation}\label{eq3.14} \int_{E_{i_{0}}} F(x, sl_{i_{0}}\zeta_{i_{0}})dx=\int_{E_{i_{0}}} F(x, s)dx. \end{equation} By (A10), \begin{equation}\label{eq3.15} \int_{D_{i_{0}}\backslash{E_{i_{0}}}} F(x, sl_{i_{0}}\zeta_{i_{0}})dx + \sum_{i\neq i_{0}}\int_{D_i} F(x, sl_i\zeta_i)dx \geq -c_ks^{2}. \end{equation} Here $c_k > 0$ depends only on $k$. Combining \eqref{eq3.11}-\eqref{eq3.15}, one has $$ I(su)\leq \frac{s^{2}\lambda^{2}_k}{2}+ c_ks^{2}-\int_{E_{i_{0}}} F(x, s)dx. $$ Substituting $s=\varepsilon_n$ and using (A11), we obtain $$ I(\varepsilon_nu)\leq \varepsilon^{2}_n\Big(\frac{s^{2}\lambda^{2}_k}{2} + c_k - (\frac{a}{2})^{2}M_n\Big). $$ Since $\varepsilon_n\to0^{+}$ and $M_n\to\infty$, we choose $n_{0}$ large enough such that the right side of the last inequality is negative. Define $$ A_k=\{\varepsilon_{n_{0}}u; u\in S_k\}. $$ Then, we have $$ \kappa(A_k)=\kappa(S_k)=k\quad \text{and}\quad \sup_{x \in A_k}I(x)<0. $$ Consequently, by Lemma \ref{lem3.3}, there exist a sequence of nontrivial critical points $(u_k)$ of $I$ such that $I(u_k)\leq 0$ for all $k\in \mathbb{N}$ and $u_k\to 0$ in $H^{\alpha}_{V}$ as $k\to \infty$. Hence, $(u_k)$ is a sequence of solutions of \eqref{eq3.3}. Therefore, for $k$ large enough, they are solutions of \eqref{eq1.1}. \end{proof} \begin{proof}[Proof of Corollary \ref{cr1.1}] By (A11'), there exist a constant $x_{0}\in \mathbb{R}^{n}$, two sequences of positives numbers $\varepsilon_n\to0$, $M_n\to \infty$ as $n\to \infty$ and constants $a_2, \varepsilon, \delta>0$ such that \begin{gather*} F(x,u)\geq \varepsilon^{2}_n M_n,\quad \text{for } |x-x_{0}|\leq\delta \text{ and } |u|= \varepsilon_n, \\ F(x,u)\geq -a_2u^{2},\quad \text{for } |x-x_{0}|\leq\delta \text{ and } |u|\leq \varepsilon, \end{gather*} which implies the condition (A11). An easy application of Theorem \ref{th1.1} shows that Corollary \ref{cr1.1} holds. This completes the proof. \end{proof} \subsection*{Acknowledgments} The author would like to thank the anonymous referees for their careful reading, critical comments, and helpful suggestions, which helped us improve the quality of this article. \begin{thebibliography}{00} \bibitem{VF} V. Ambrosio, G. M. 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