\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 88, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/88\hfil Fractional Schr\"odinger equations] {Infinitely many solutions for fractional Schr\"odinger equations in $\mathbb{R}^N$} \author[C. Chen \hfil EJDE-2016/88\hfilneg] {Caisheng Chen} \address{Caisheng Chen \newline College of Science, Hohai University, Nanjing 210098, China} \email{cshengchen@hhu.edu.cn} \thanks{Submitted January 31, 2016. Published March 30, 2016.} \subjclass[2010]{35R11, 35A15, 35J60, 47G20} \keywords{ Fractional Schr\"odinger equation; variational methods; \hfill\break\indent (PS) condition; (C)$_c$ condition} \begin{abstract} Using variational methods we prove the existence of infinitely many solutions to the fractional Schr\"odinger equation \[ (-\Delta)^su+V(x)u=f(x,u), \quad x\in\mathbb{R}^N, \] where $N\ge 2, s\in (0,1)$. $(-\Delta)^s$ stands for the fractional Laplacian. The potential function satisfies $V(x)\ge V_0>0$. The nonlinearity $f(x,u)$ is superlinear, has subcritical growth in $u$, and may or may not satisfy the (AR) condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction and statement of main results} In this article, we investigate the existence of infinitely many solutions to the fractional Schr\"odinger equation \begin{equation}\label{1.1} (-\Delta)^su+V(x)u=f(x,u), \quad x\in\mathbb{R}^N, \end{equation} where $N\ge 2$, $s\in (0,1)$. $(-\Delta)^s$ stands for the fractional Laplacian. The function $f(x,u)$ is odd, sublinear or suplinear and subcritical in $u$, $V(x)$ is positive and bounded below in $\mathbb{R}^N$. Equation \eqref{1.1} arises in the study of the fractional Schr\"odinger equation \begin{equation}\label{1.2} i\frac{\partial \psi}{\partial t}+(-\Delta)^s\psi+V(x)\psi=f(x,\psi), \quad x\in\mathbb{R}^N,\;t>0, \end{equation} when looking for standing waves, that is, solutions with the form $\psi(x,t)=e^{i\omega t}u(x),$ where $\omega$ is a constant. This equation was introduced by Laskin \cite{F15,F16} and comes from an expansion of the Feynman path integral and from Brownian-like to L\'{e}vy-like quantum mechanical paths. This equation is of particular interest in fractional quantum mechanics for the study of particles on stochastic fields modelled by L\'{e}vy processes, which occur widely in physics, chemistry and biology. The stable L\'{e}vy processes that gives rise to equations with the fractional Laplacian have recently attracted much research interest. For more details, we can see \cite{F5}. Nonlinear equations like \eqref{1.1} have recently been studied by Cabr\'e and Roquejoffre \cite{F3}, Cabr\'e and Tan \cite{F4}, Sire and Valdinoci \cite{F23}, Iannizzotto et al. \cite{F13}, Hua and Yu \cite{F12}. A one-dimensional version of \eqref{1.1} has been studied in the context of solitary waves by Weinstein \cite{F28}. Equations of the form \eqref{1.1} in the whole space $\mathbb{R}^N$ were studied by a number of authors; see for instance \cite{F6,F9,F20,F21,F22} and the references therein. Felmer et al. \cite{F9} considered the existence and regularity of positive solution of \eqref{1.1} with $V(x)=1$ and $s\in(0,1)$ when $f$ has subcritical growth and satisfies the Ambrosetti-Rabinowitz ((AR) for short) condition. Secchi \cite{F20} obtained the existence of ground state solutions of \eqref{1.1} for $s\in(0,1)$ when $V(x)\to \infty$ as $|x|\to \infty$ and (AR) condition holds. In \cite{F8}, the authors proved the existence of infinitely many weak solutions for \eqref{1.1} by variant fountain theorem under the assumption \begin{equation}\label{1.3} 0<\inf_{x\in\mathbb{R}^N}V(x)<\liminf_{|x|\to\infty}V(x)=V_{\infty}<\infty. \end{equation} Tang \cite{F26} studied \eqref{1.1} with a potential $V(x)$ satisfying \begin{equation}\label{1.4} 0<\inf_{x\in\mathbb{R}^N}V(x),\quad \operatorname{meas}(\{x\in \mathbb{R}^N |V(x)\le d\})<\infty,\quad \forall d>0. \end{equation} Similar assumptions can be found in \cite{F10,F22,F24,F29}. Each of these conditions ensures that the embedding $W^{s,2}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)$ is compact for $2\le q<2_s^*=\frac{2N}{N-2s}$. On the other hand, Gou and Sun \cite{F11}, Chang and Wang \cite{F7} investigated the existence of radial solutions for \eqref{1.1}. In this article, we are interest in the existence of infinitely many solutions for \eqref{1.1} under the assumptions (A3)--(A7) below. Our assumptions on $f(x,u)$ are different from that in the above papers. The weighted functions $h_1(x),h_2(x)$ and $h_3(x)$ depend on the potential function $V(x)$ and the nonlinear function $f(x,u)$ either satisfies (AR) condition or does not. Moreover, two cases that $f(x,u)$ is bounded and unbounded in $x\in\mathbb{R}^N$ are considered. We note that, in \cite{F18,F24,F27}, $f(x,u)$ is assumed to bounded in $x\in\mathbb{R}^N$ To state our main results, we recall some fractional Sobolev spaces and norms \cite{F17}. Let $V(x)$ satisfy (A1) below and \begin{equation}\label{1.5} E=\Big\{u\in W^{s,2}(\mathbb{R}^N): \int_{\mathbb{R}^N} |\xi|^{2s}| \hat{u}|^2d\xi+\int_{\mathbb{R}^N} V(x)|u|^2dx<\infty\Big\} \end{equation} endowed with the norm \begin{equation}\label{1.6} \|u\|_{E}=\Big(\int_{\mathbb{R}^N} |\xi|^{2s}| \hat{u}|^2d\xi+\|u\|_{2,V}^2\Big)^{1/2}, \end{equation} where and in the sequel, $\|u\|_{2,V}^2=\int_{\mathbb{R}^N} V(x)|u|^2dx$ and $\hat{\omega}=\hat{\omega}(\xi)$ is the Fourier transform of $\omega(x)$; that is, \begin{equation}\label{1.7} \begin{gathered} \hat{\omega}=\mathcal{F}[\omega(x)] =\frac{1}{(2\pi)^{N/2}}\int_{\mathbb{R}^N}\omega(x)e^{-i\xi\cdot x}dx,\\ \omega(x)=\mathcal{F}^{-1}[\hat{\omega}] =\frac{1}{(2\pi)^{N/2}}\int_{\mathbb{R}^N} \hat{\omega}(\xi)e^{i\xi\cdot x}d\xi. \end{gathered} \end{equation} In \cite{F17}, the author shows that \begin{gather}\label{1.8} ((-\Delta)^su)(x)=\mathcal{F}^{-1}[|\xi|^2\hat{u}],\quad \forall x\in\mathbb{R}^N, \\ \label{1.9} [u]^2_{E}=\frac{2}{C(N,s)}\int_{\mathbb{R}^N} |\xi|^2|\hat{u}|^2d\xi, \end{gather} where \begin{equation}\label{1.10} [u]_{E}=\Big(\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy \Big)^{1/2} \end{equation} is called the Gagliardo norm, and the constant $C(N,s)$ depends only on the space dimensional $N$ and the order $s$, and it is explicitly given by the integral \begin{equation}\label{1.11} \frac{1}{C(N,s)}=\int_{\mathbb{R}^N}\frac{1-\cos(\zeta_1)}{|\zeta|^{N+2s}}d\zeta, \quad \zeta=(\zeta_1,\zeta_2,\dots,\zeta_N)\in\mathbb{R}^N. \end{equation} Moreover, by the Plancherel formula in Fourier analysis, we have \begin{equation}\label{1.12} [u]^2_{E}=\frac{2}{C(N,s)}\|(-\Delta)^{s/2}u\|_2^2. \end{equation} Then, from \eqref{1.8}-\eqref{1.12}, we obtain that the norm $\|\cdot\|_E$ is equivalent to the norms \begin{equation}\label{1.13} \begin{gathered} \|u\|_{1}=\Big(\int_{\mathbb{R}^N} V(x)|u|^2dx +\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Big)^{1/2}, \\ \|u\|_{2}=\Big(\int_{\mathbb{R}^N} V(x)|u|^2dx+\|(-\Delta)^{s/2}u\|^2_2\Big)^{1/2}. \end{gathered} \end{equation} In general, we define the fractional Sobolev space $W^{s,p}(\mathbb{R}^N)(00$ in $\mathbb{R}^N$. Then, for any $u\in E$, \begin{equation}\label{1.20} \|u\|_{q}\le S_q\|u\|_E \end{equation} where $S_q$ is a constant depending on $s,q,p,N$ and $V_0$. In particular, we denote $S_{2_s^*}$ by $S_0$. \end{lemma} \begin{definition} \label{def1.1} \rm A function $u\in E$ is said to be a (weak) solution of \eqref{1.1} if for any $\varphi \in E$, we have \begin{equation}\label{1.21} \int_{\mathbb{R}^N} |\xi|^{2s}\hat{u}\hat{\varphi} d\xi+\int_{\mathbb{R}^N} V(x)u\varphi dx = \int_{\mathbb{R}^N} f(x,u)\varphi dx. \end{equation} \end{definition} Let $J(u):E\to \mathbb{R} $ be the energy functional associated with \eqref{1.1} defined by \begin{equation}\label{1.22} J(u)=\frac{1}{2}\int_{\mathbb{R}^N} |\xi|^{2s}|\hat{u}|^2 d\xi +\frac{1}{2}\int_{\mathbb{R}^N} V(x)|u|^2dx -\int_{\mathbb{R}^N} F(x,u)dx, \end{equation} where $F(x,u)=\int_0^u f(x,t)dt$. Using \eqref{1.18} and assumptions (A3)--(A7) below, we see that the functional $J$ is well defined and $J\in C^1(E,\mathbb{R})$ with \begin{equation}\label{1.23} J'(u)\varphi =\int_{\mathbb{R}^N} |\xi|^{2}\hat{u}\hat{\varphi} d\xi+\int_{\mathbb{R}^N} V(x)u\varphi dx -\int_{\mathbb{R}^N} f(x,u)\varphi dx, \quad \forall \varphi\in E. \end{equation} Throughout this article, the function $f(x,u)\in C(\mathbb{R}^N \times\mathbb{R})$ is odd in $u$. In addition, we use the following assumptions. \begin{itemize} \item[(A1)] The function $V(x)\in C(\mathbb{R}^N)$ satisfies $\inf_{x\in\mathbb{R}^N}V(x)\ge V_0>0$, where $V_0$ is a constant. \item[(A2)] There exists $a>0$ such that $\lim_{|y|\to\infty}\operatorname{meas}(\{x\in B_a(y):V(x)\le d\})=0$ for any $d>0$, where ``meas" denotes the Lebesgue measure on $\mathbb{R}^N$ and $B_r(x)$ denotes any open ball of $\mathbb{R}^N$ centered at $x$ and of radius $r>0$, while we simply write $B_r$ when $x=0$. \item[(A3)] There exist $2<\alpha<\beta<2_s^*$ such that \begin{equation}\label{1.24} |f(x,u)|\le h_1(x)|u|^{\alpha-1}+h_2(x)|u|^{\beta-1},\forall (x,u)\in \mathbb{R}^N \times\mathbb{R}, \end{equation} where $h_1(x), h_2(x)\in C(\mathbb{R}^N)$ and \begin{equation}\label{1.25} \lim_{r\to\infty}\sup_{x\in B_r^c}\frac{h_1(x)}{V^{t_1}(x)}=0,\quad \lim_{r\to\infty}\sup_{x\in B_r^c} \frac{h_2(x)}{V^{t_2}(x)}=0 \end{equation} with $t_1=(2_s^*-\alpha)/(2_s^*-2),\; t_2=(2_s^*-\beta)/(2_s^*-2)$ and $B_r^c=\mathbb{R}^N\setminus\overline{B}_r=\{x\in \mathbb{R}^N: |x|>r\}$. \item[(A4)] There exists $\mu>2$ such that \begin{equation}\label{1.26} tf(x,t)-\mu F(x,t)\ge 0,\quad \forall (x,t )\in \mathbb{R}^N\times \mathbb{R}. \end{equation} \item[(A5)] $\lim_{|t|\to\infty}(F(x,t)t^{-2})=\infty$ for any $x\in \mathbb{R}^N$. \item[(A6)] There exist $k>\frac{N}{2s}$ and $2<\alpha<\beta\le\frac{2k}{k-1}$ such that \eqref{1.24} and \eqref{1.25} hold. Furthermore, there exist $b, c_1\ge 1$ such that for $x\in\mathbb{R}^N$ and $|u|\ge b$, \begin{equation}\label{1.27} \begin{gathered} F(x,u)\ge 0,\quad G(x,u) = \frac{1}{2}uf(x,u) - F(x,u)\ge 0,\\ |F(x,u)|^{k}\le c_1^k|u|^{2k}|h_3(x)|^{2k}G(x,u), \end{gathered} \end{equation} where $h_3(x)\in C(\mathbb{R}^N)$ satisfies \begin{equation}\label{1.28} \lim_{r\to\infty}\sup_{x\in B_r^c}\frac{|h_3(x)|^{2k'}}{V^{t_3}(x)}=0, \quad\text{with } k'=\frac{k}{k-1},\; t_3=\frac{2_s^*-2k'}{2_s^*-2}. \end{equation} \item[(A7)] There exist a constant $C_0>0$ and $2<\alpha<\beta<2_s^*$, such that \begin{equation}\label{1.29} |f(x,u)|\le C_0(|u|^{\alpha-1}+|u|^{\beta-1}),\quad \forall (x,u)\in \mathbb{R}^N \times\mathbb{R}. \end{equation} \end{itemize} \begin{remark} \label{rmk1.5} \rm Condition (A2), which is weaker than the coercivity assumption $V(x)\to\infty$ as $|x|\to\infty$, was originally introduced by Bartsch and Wang in \cite{F1} to overcome the lack of compactness. Clearly, if $V(x)\to\infty$ as $|x|\to\infty$, it is possible that the functions $h_1(x),h_2(x)$ and $h_3(x)$ in $(A3)$ and (A6) are unbounded on $\mathbb{R}^N$. So, it is necessary to consider the condition (A7). \end{remark} Our main results in this paper are as follows. \begin{theorem} \label{thm1.6} Let $s\in(0,1), 2s\frac{N}{2s}$ in (A7) implies that $\frac{2k}{k-1}<2_s^*$. \end{remark} \begin{remark} \label{rmk1.9} \rm Assumption (A4) is called the (AR) condition. Obviously, the power functions in $u$ like $f(x,u)=\sum_{i=1}^nh_i(x)|u|^{\beta_i-2}u$ with $2<\beta_i<2_s^*$ satisfy (A3) and (A4) for appropriate functions $h_i\in C(\mathbb{R}^N)$. The functions like $f(x,u)=h(x)u\log(1+|u|)$ fails to satisfy condition (A4), but it satisfies (A6). \end{remark} Teng \cite{F27} considered problem \eqref{1.1} under assumption (A4) with $h_1(x), h_2(x)\in L^{\infty}(\mathbb{R}^N)$. Obviously, our assumptions on $h,h_1$ and $h_2$ are weaker than that in \cite{F27}. Without loss of generality, we let $V_0=1$ in (A1). \section{Proof of main results} To prove the main results, we recall some useful concepts and results. \begin{definition} \label{def2.1} \rm Let $E$ be a real Banach space and the functional $J\in C^1(E,\mathbb{R})$. We say that $J$ satisfies the $(C)_c$ condition if any $(C)_c$ sequence $\{u_n\}\subset E$: \begin{equation}\label{2.1} J(u_n)\to c,\quad (1+\|u_n\|_E)\|J'(u_n)\|_{E^*}\to 0 \quad \text{as } n\to \infty \end{equation} has a convergent subsequence in $E$. \end{definition} \begin{lemma}[\cite{F19,F25}] \label{lem2.1} Let $E$ be an infinite dimensional real Banach space, the functional $J\in C^1(E,\mathbb{R})$ be even and satisfy the $(C)_c$ condition for all $c>0$ and $J(0)=0$. In addition, assume $E=Y\oplus Z$, in which $Y$ is finite dimensional, and $J$ satisfies \begin{itemize} \item[(A8)] there exist constants $\rho, \alpha_0>0$ such that $J(z)\ge \alpha_0$ on $\partial B_{\rho}\cap Z$; \item[(A9)] for each finite dimensional subspace $E_0\subset E$, there is an $R=R(E_0)$ such that $J(z)\le 0$ on $E_0\setminus \overline{B}_{R}$, where $B_R=\{z\in E: \|z\|_E2$. Then for large $n$, we have \begin{equation}\label{2.2} \begin{aligned} c+1+\|u_n\|_E &\ge J(u_n) - \frac{1}{\mu}J'(u_n)u_n\\ &=(\frac{1}{2}-\frac{1}{\mu})\|u_n\|_E^2+\frac{1}{\mu}\int_{\mathbb{R}^N}f(x,u_n)u_n -\mu F(x,u_n))dx. \end{aligned} \end{equation} Then (A4) implies that $\{u_n\}$ is bounded in $E$. The proof is complete. \end{proof} \begin{lemma} \label{lem2.4} Assume {\rm (A1), (A2), (A5), (A6)} hold. Then any $(C)_c$ sequence $\{u_n\}$ is bounded in $E$. \end{lemma} \begin{proof} To prove the boundedness of $\{u_n\}$, arguing contradiction, we suppose that $\|u_n\|_E\to \infty$ as $n\to\infty$. Let $v_n(x)=\frac{u_n(x)}{\|u_n\|_E}$. Then $\|v_n\|_E=1$ for all $n\ge 1$. By Lemma \ref{lem2.2}, there exists a subsequence of $\{v_n\}$, still denoted by $\{v_n\}$, and $v\in E$ such that $\|v\|_E\le 1$ and \begin{equation}\label{2.3} \begin{gathered} v_n \rightharpoonup v \text{ weakly in }E; \quad v_n\to v\text{ in } L^q(\mathbb{R}^N)\; (2\le q<2_s^*);\\ v_n(x)\to v(x) \text{ a.e. in } \mathbb{R}^N. \end{gathered} \end{equation} Clearly, it follows from \eqref{2.3} that there exists $\omega(x)\in L^q(\mathbb{R}^N)(2\le q<2_s^*)$ such that $|v_n(x)|\le \omega(x)$ a.e. in $\mathbb{R}^N$ for all $n\ge 1$. From \eqref{1.22}, \eqref{1.23} and \eqref{2.1}, it follows that for, $n$ large, \begin{equation}\label{2.4} c+ 1 \ge J(u_n) - \frac{1}{2}J'(u_n)u_n = \int_{\mathbb{R}^N} G(x,u_n)dx, \end{equation} where $ G(x,u)=\frac{1}{2}uf(x,u)-F(x,u)$, and \begin{equation}\label{2.5} \begin{aligned} \frac{1}{2} &\leq \limsup_{n\to\infty}\int_{\mathbb{R}^N} \frac{|F(x,u_n)|}{\|u_n\|_E^{2}}dx\\ &\leq \limsup_{n\to\infty}\int_{B_r} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}dx +\limsup_{n\to\infty}\int_{B_r^c} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}dx, \end{aligned} \end{equation} for any $r>0$. By (A6), we obtain, for any $\varepsilon>0$, there exists $\delta>0$ such that $\frac{|F(x,t)|}{|t|^{2}}\le \varepsilon(h_1(x)+h_2(x))$ for all $0<|t|\le\delta$ and all $x\in\mathbb{R}^N$. Denote $X_n= \{x\in\mathbb{R}^N:|u_n(x)|\le \delta\}$, $Y_n= \{x\in\mathbb{R}^N:\delta<|u_n(x)|\le b\}, Z_n =\{x\in\mathbb{R}^N:|u_n(x)|\ge b\}$, where the constant $b$ is given in (A6). Obviously, $\mathbb{R}^N=X_n\cup Y_n \cup Z_n$ and $B_r^c=B_r^c\cap( X_n\cup Y_n \cup Z_n)$. Then \begin{equation}\label{2.6} \begin{split} \int_{B_r^c\cap X_n}\frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}dx &\le \varepsilon \int_{B_r^c\cap X_n} (h_1(x)+h_2(x))|v_n|^{2}dx \\ &\le \varepsilon (\|h_1V^{-1}\|_{\infty}+\|h_2V^{-1}\|_{\infty}) \int_{\mathbb{R}^N} V|v_n|^{2}dx \\ &\le 2\varepsilon M_1\|v_n\|_E^2=2\varepsilon M_1, \end{split} \end{equation} where $M_1=\max\{\|h_1V^{-1}\|_{\infty},\|h_2V^{-1}\|_{\infty}\}$. Furthermore, by \eqref{1.24} and \eqref{1.25}, one sees that \begin{equation}\label{2.7} \begin{split} &\int_{B_{r}^c\cap Y_n} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2 \,dx\\ &\le b^{\beta-2}\int_{B_{r}^c\cap Y_n} (h_1(x)+h_2(x))|v_n(x)|^2 \,dx \\ &\le b^{\beta-2}\Big(\sup_{x\in B_r^c}\frac{h_1(x)}{V(x)}+\sup_{x\in B_r^c}\frac{h_2(x)}{V(x)} \Big)\int_{B_r^c}V(x)|v_n|^2dx \\ &\le b^{\beta-2}\Big(\sup_{x\in B_r^c}\frac{h_1(x)}{V(x)}+\sup_{x\in B_r^c}\frac{h_2(x)}{V(x)} \Big)\to 0 \quad \text{as } r=|x|\to \infty. \end{split} \end{equation} On the other hand, from \eqref{1.27} and \eqref{2.4} it follows that \begin{equation}\label{2.8} \begin{split} &\int_{B_r^c\cap Z_n} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx\\ &\le\Big(\int_{B_r^c\cap Z_n} \Big(\frac{|F(x,u_n)|}{h_3^2|u_n|^2}\Big)^{k} dx\Big)^{1/k}\Big(\int_{B_r^c\cap Z_n} |h_3v_n|^{2k'}dx\Big)^{1/k'} \\ &\le c_1\Big(\int_{\mathbb{R}^N} G(x,u_n)dx\Big)^{1/k}\Big(\int_{B_r^c\cap Z_n} |h_3v_n|^{2k'}dx\Big)^{1/k'}\\ &\le c_1(c+1)^{1/k}\Big(\int_{B_r^c} |h_3v_n|^{2k'}dx\Big)^{1/k'}. \end{split} \end{equation} Note that $2<2k'<2_s^*$. Let $t_3=(2_s^*-2k')/(2_s^*-2)$. By the H\"older inequality and \eqref{1.20}, we obtain \begin{equation}\label{2.9} \begin{aligned} \int_{B_r^c} |h_3v_n|^{2k'}dx &\le \sup_{x\in B_r^c}\frac{|h_3(x)|^{2k'}}{V^{t_3}(x)} \Big(\int_{B_r^c} Vv_n^2dx\Big)^{t_3} \Big(\int_{B_r^c} |v_n|^{2_s^*}dx\Big)^{1-t_3} \\ &\le S_0\|v_n\|_E^{2k'}\sup_{x\in B_r^c}\frac{|h_3(x)|^{2k'}}{V^{t_3}(x)} \le S_0\sup_{x\in B_r^c}\frac{|h_3(x)|^{2k'}}{V^{t_3}(x)}\to 0 \end{aligned} \end{equation} as $r=|x|\to\infty$, where $S_0=S_{2_s^*}$ is the constant in \eqref{1.20}. Then, and application of \eqref{2.6}-\eqref{2.9} implies that for any $\varepsilon>0$, there exist $n_0, r_0\ge 1$, such that $n\ge n_0$, $r\ge r_0$, \begin{equation}\label{2.10} \int_{B_r^c} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx\le \varepsilon(2M_1+1). \end{equation} Set $T_n=X_n\cup Y_n$. Notice that for all $x\in B_{r_0}\cap T_n$, \begin{equation}\label{2.11} \begin{aligned} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2} & \le b^{\beta-2}(h_1(x)+h_2(x))|v_n(x)|^2\\ &\le b^{\beta-2}M_2|\omega(x)|^2\equiv d(x)\in L^1(B_{r_0}), \end{aligned} \end{equation} where \begin{equation}\label{2.12} M_2=\sup_{x\in B_{r_0}}(h_1(x)+h_2(x)). \end{equation} If $v(x)=0$ in $B_{r_0}$, it follows from Fatou's lemma that \begin{equation}\label{2.13} \begin{aligned} \limsup_{n\to\infty}\int_{B_{r_0}\cap T_n} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}dx &\le M_2b^{\beta-2} \int_{B_{r_0}} \limsup_{n\to\infty}|v_n|^{2}dx\\ &=M_2b^{\beta-2} \int_{B_{r_0}} |v|^{2}dx=0. \end{aligned} \end{equation} Arguing as in \eqref{2.8} and \eqref{2.9}, we obtain \begin{equation}\label{2.14} \int_{B_{r_0}\cap Z_n} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx \le C_1\sup_{x\in B_{r_0}}|h_3(x)|^2\Big(\int_{B_{r_0}} |v_n|^{2k'}dx\Big)^{1/k'} \end{equation} with $C_1=c_1(c+1)^{1/k}$. Similarly, since $|v_n(x)|^{2k'}\le |\omega(x)|^{2k'}$ a.e. in $\mathbb{R}^N$ and $|\omega(x)|^{2k'} \in L^1(\mathbb{R}^N)$, we obtain \begin{equation*} \limsup_{n\to\infty}\int_{B_{r_0}} |v_n|^{2k'}dx\le \int_{B_{r_0}} \limsup_{n\to\infty}|v_n|^{2k'}dx=\int_{B_{r_0}} |v|^{2k'}dx=0. \end{equation*} Hence, \begin{equation}\label{2.15} \limsup_{n\to\infty}\int_{B_{r_0}} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx=0. \end{equation} So, an application of \eqref{2.10} and \eqref{2.15} contradicts \eqref{2.5} and then meas$(A)>0$, where $A=\{x\in\mathbb{R}^N:v(x)\neq 0\}$. Obviously, for a.e. $x\in A$, we have $\lim_{n\to\infty}|u_n(x)|=\infty$. Hence, $A\subset Z_n$ for large $n$. Moreover, one sees that \begin{equation}\label{2.16} \begin{split} &\int_{T_n} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx\le b^{\beta-2}\int_{\mathbb{R}^N}(h_1(x)+h_2(x))|v_n|^2dx \\ &\le b^{\beta-2}(\|h_1V^{-1}\|_{\infty}+\|h_2V^{-1}\|_{\infty}) \int_{\mathbb{R}^N} V|v_n|^2dx\\ &\le 2b^{\beta-2}M_1\|v_n\|_E^2=2b^{\beta-2}M_1. \end{split} \end{equation} Moreover, using assumption (A5) and Fatou's lemma, it follows from $J(u_n)\to c$ that \begin{equation}\label{2.17} \begin{split} 0&=\lim_{n\to\infty}\frac{c+o(1)}{\|u_n\|_E^{2}}=\lim_{n\to\infty}\frac{J(u_n)}{\|u_n\|_E^{2}}\le \lim_{n\to\infty}\Big[\frac{1}{2}-\int_{\mathbb{R}^N} \frac{F(x,u_n)}{|u_n|^{2}}|v_n|^{2}dx\Big] \\ &\le \frac{1}{2}+2b^{\beta-2}M_1-\liminf_{n\to\infty}\int_{Z_n} \frac{F(x,u_n)}{|u_n|^2}|v_n|^2dx \\ &\le \frac{1}{2}+2b^{\beta-2}M_1-\int_{\mathbb{R}^N} \liminf_{n\to\infty}\frac{F(x,u_n)}{|u_n|^{2}}\chi_{Z_n}(x)|v_n|^{2}dx =-\infty, \end{split} \end{equation} where $\chi_I$ denotes the characteristic function associated to the mensurable subset $I\subset \mathbb{R}^N$. Clearly, \eqref{2.17} is impossible. Thus $\{u_n\}$ is bounded in $E$ and the proof of Lemma \ref{lem2.4} is finished. \end{proof} \begin{lemma} \label{lem2.5} Assume {\rm (A1), (A2), (A5), (A7)}. In addition, suppose that \eqref{1.27} is satisfied with $h_3(x)\equiv 1$. Then any $(C)_c$ sequence $\{u_n\}$ is bounded in $E$. \end{lemma} \begin{proof} Arguing as the proof of Lemma \ref{lem2.4}, we suppose that $\|u_n\|_E\to \infty$ as $n\to\infty$. Let $v_n(x)=\frac{u_n(x)}{\|u_n\|_E}$. Then $\{v_n\}$ satisfies \eqref{2.3}. For any $\varepsilon>0$, we choose $r_1>0$ such that $\int_{B_{r}^c}|v(x)|^2dx<\varepsilon$ when $r\ge r_1$. Since $v_n(x)\to v(x)$ in $L^2(\mathbb{R}^N)$, we obtain \begin{equation}\label{2.18} \limsup_{n\to\infty}\int_{B_r^c}|v_n(x)|^2dx\le \int_{B_r^c}\limsup_{n\to\infty}|v_n(x)|^2dx \le\int_{B_{r}^c}|v(x)|^2dx<\varepsilon. \end{equation} Then \begin{equation}\label{2.19} \begin{aligned} \limsup_{n\to\infty}\int_{B_r^c\cap T_n} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}dx &\le 2b^{\beta-2}C_0 \limsup_{n\to\infty}\int_{B_r^c} |v_n|^{2}dx\\ &\le C_2 \int_{B_r^c} |\omega(x)|^{2}dx\le C_2\varepsilon, \end{aligned} \end{equation} where $C_2=2b^{\beta-2}C_0$, $b$ is the constant in (A6) and $C_0$ is given in \eqref{1.29}. On the other hand, from \eqref{1.27} with $h_3(x)=1$ and \eqref{2.4} it follows that \begin{equation}\label{2.20} \begin{split} &\int_{B_r^c\cap Z_n} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx \\ & \le\Big(\int_{B_r^c\cap Z_n} \Big(\frac{|F(x,u_n)|}{|u_n|^2}\Big)^{k}dx\Big)^{1/k} \Big(\int_{B_r^c\cap Z_n} |v_n|^{2k'}dx\Big)^{1/k'} \\ &\le c_1\Big(\int_{\mathbb{R}^N} G(x,u_n)dx\Big)^{1/k}\Big(\int_{B_r^c\cap Z_n} |v_n|^{2k'}dx\Big)^{1/k'}\\ &\le c_1(c+1)^{1/k}\Big(\int_{B_r^c} |v_n|^{2k'}dx\Big)^{1/k'}. \end{split} \end{equation} From \eqref{2.9} and \eqref{2.18}, for large $n$, we obtain \begin{equation}\label{2.21} \int_{B_r^c} |v_n|^{2k'}dx\le \|v_n\|_{L^2(B_r^c)}^{2t_3}\|v_n\|_{L^{2_s^*}(B_r^c)}^{(1-t_3)2^*_s}\le S_0\|v_n\|_{L^2(B_r^c)}^{2t_3}\le S_0\varepsilon. \end{equation} Then, an application of \eqref{2.19}-\eqref{2.21} gives that for any $\varepsilon>0$ there exist $n_0, r_0\ge 1$ such that $n\ge n_0, r\ge r_0$, \begin{equation}\label{2.22} \int_{B_r^c} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx\le \varepsilon(C_2+S_0C_1). \end{equation} Similar to \eqref{2.11}, for a.e. $x\in B_{r_0}\cap T_n $ and $n\ge 1$, we obtain \begin{equation}\label{2.23} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}\le C_2|v_n(x)|^2\le C_2|\omega(x)|^2\equiv d(x)\in L^1(B_{r_0}). \end{equation} By Fatou's lemma, \begin{equation}\label{2.24} \begin{aligned} \limsup_{n\to\infty}\int_{B_{r_0}\cap T_n} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2} & \le C_2 \int_{B_{r_0}}\limsup_{n\to\infty}|v_n(x)|^2\\ &= C_2\int_{B_{r_0}}|v(x)|^2dx. \end{aligned} \end{equation} Similar to \eqref{2.20}, we derive \begin{equation}\label{2.25} \limsup_{n\to\infty}\int_{B_{r_0}\cap Z_n} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}\le C_1\Big(\int_{B_{r_0}} |v(x)|^{2k'}dx\Big)^{1/k'}. \end{equation} If $v(x)=0$ in $B_{r_0}$, an application of \eqref{2.24} and \eqref{2.25} gives that \begin{equation}\label{2.26} \limsup_{n\to\infty}\int_{B_{r_0}} \frac{|F(x,u_n)|}{|u_n|^{2}}|v_n|^{2}=0. \end{equation} Combining \eqref{2.22} with \eqref{2.26} contradicts \eqref{2.5}. So, $\operatorname{meas}(A)>0$, where $A=\{x\in\mathbb{R}^N:v(x)\not=0\}$ and for a.e. $x\in A$, we have $\lim_{n\to\infty}|u_n(x)|=\infty$. Hence, $A\subset Z_n$ for large $n$. Moreover, one sees that \begin{equation}\label{2.27} \int_{T_n} \frac{|F(x,u_n)|}{|u_n|^2}|v_n|^2dx\le C_1 \int_{\mathbb{R}^N}|v_n|^2dx\le C_1\int_{\mathbb{R}^N} V|v_n|^2dx\le C_1\|v_n\|_E^2=C_1. \end{equation} Moreover, using assumption (A5) and Fatou's lemma, from $J(u_n)\to c$ it follows that \begin{equation}\label{2.28} \begin{split} 0&=\lim_{n\to\infty}\frac{c+o(1)}{\|u_n\|_E^{2}} =\lim_{n\to\infty}\frac{J(u_n)}{\|u_n\|_E^{2}}\le \lim_{n\to\infty}\Big[\frac{1}{2}-\int_{\mathbb{R}^N} \frac{F(x,u_n)}{|u_n|^{2}}|v_n|^{2}dx\Big] \\ &\le \frac{1}{2}+C_1-\liminf_{n\to\infty}\int_{Z_n} \frac{F(x,u_n)}{|u_n|^2}|v_n|^2dx \\ &\le \frac{1}{2}+C_1-\int_{\mathbb{R}^N} \liminf_{n\to\infty} \frac{F(x,u_n)}{|u_n|^{2}}\chi_{Z_n}(x)|v_n|^{2}dx =-\infty. \end{split} \end{equation} Clearly, the limit \eqref{2.28} is impossible. Thus $\{u_n\}$ is bounded in $E$ and the proof is complete. \end{proof} From Lemmas \ref{lem2.3}--\ref{lem2.5}, we know that any $(PS)_c$ sequence and $(C)_c$ sequence $\{u_n\}$ of the functional $J$ are bounded in $E$. Therefore, by Lemma \ref{lem2.2}, there exists a subsequence of $\{u_n\}$, still denoted by $\{u_n\}$, and $u\in E$ such that $\|u_n\|_E+\|u\|_E\le M(\forall n\ge 1)$ and \begin{equation}\label{2.29} \begin{gathered} u_n \rightharpoonup u \text{ weakly\ in } E, \quad u_n\to u\text{ in } L^q(\mathbb{R}^N)\; (2\le q<2_s^*),\\ u_n(x)\to u(x) \text{ a.e. in } \mathbb{R}^N \end{gathered} \end{equation} with some constant $M>0$. \begin{lemma} \label{lem2.6} Assume {\rm (A1)--(A6)} hold. If the sequence $\{u_n\}$ satisfies \eqref{2.29}, then \begin{gather}\label{2.30} \lim_{n\to\infty}\int_{\mathbb{R}^N} h_1(|u_n|^{\alpha}-|u|^{\alpha})dx=0,\quad \lim_{n\to\infty}\int_{\mathbb{R}^N} h_2(|u_n|^{\beta}-|u|^{\beta})dx=0,\\\ \label{2.31} \lim_{n\to\infty}\int_{\mathbb{R}^N} f(x,u_n)(u_n-u)dx=0,\quad \lim_{n\to\infty}\int_{\mathbb{R}^N} f(x,u)(u_n-u)dx=0, \\ \label{2.32} \lim_{n\to\infty}\int_{\mathbb{R}^N} F(x,u_n)dx=\int_{\mathbb{R}^N} F(x,u)dx. \end{gather} \end{lemma} \begin{proof} First, we assume {(A3), (A4)}. From \eqref{2.29}, we obtain \begin{equation}\label{2.33} \lim_{n\to\infty}\int_{B_r} h_1(|u_n|^{\alpha}-|u|^{\alpha})dx=0,\quad \lim_{n\to\infty}\int_{B_r} h_2(|u_n|^{\beta}-|u|^{\beta})dx=0 \end{equation} for any $r>0$. On the other hand, we see from the H\"older inequality and (A3) that \begin{equation}\label{2.34} \begin{split} \int_{B^c_{r}} h_1|u_n|^{\alpha}dx &\le \sup_{x\in B_r^c}\frac{h_1(x)}{V^{t_1}(x)} \Big(\int_{B^c_{r}} V|u_n|^2dx\Big)^{t_1} \Big(\int_{B_r^c} |u_n|^{2_s^*}dx\Big)^{1-t_1} \\ &\le S_0\sup_{x\in B_r^c}\frac{h_1(x)}{V^{t_1}(x)}\|u_n\|_E^{2t_1}\|u_n\|_E^{(1-t_1)2_s^*} \\ &\le S_0 M^{\alpha} \sup_{x\in B_r^c}\frac{h_1(x)}{V^{t_1}(x)} \to 0, \quad \text{as } r\to\infty, \end{split} \end{equation} where $t_1=(2_s^*-\alpha)/(2_s^*-2), S_0=S_{2_s^*}$. Similarly, as $ r\to \infty$, \begin{equation}\label{2.35} \begin{aligned} \int_{B^c_{r}} h_2|u_n|^{\beta}dx &\le S_0\sup_{x\in B_r^c}\frac{h_2(x)}{V^{t_2}(x)} \|u_n\|_E^{2t_2}\|u_n\|_E^{(1-t_2)2_s^*} \\ &\le S_0 M^{\beta} \sup_{x\in B_r^c}\frac{h_2(x)}{V^{t_2}(x)} \to 0, \end{aligned} \end{equation} where $t_2=(2_s^*-\beta)/(2_s^*-2)$. Then an application of \eqref{2.33}, \eqref{2.34} and \eqref{2.35} gives \eqref{2.30}. Moreover, the limit \eqref{2.30} and Brezis-Lieb lemma \cite{F2} give that \begin{equation}\label{2.36} \lim_{n\to\infty}\int_{\mathbb{R}^N} h_1|u_n-u|^{\alpha}dx=0,\quad \lim_{n\to\infty}\int_{\mathbb{R}^N} h_2|u_n-u|^{\beta}dx=0. \end{equation} Thus, from \eqref{2.36}, it follows that \begin{equation}\label{2.37} \begin{split} &\int_{\mathbb{R}^N} h_1|u_n|^{\alpha-1}|u_n-u|dx\le \Big(\int_{\mathbb{R}^N} h_1|u_n|^{\alpha}dx\Big)^{(\alpha-1)/\alpha} \Big(\int_{\mathbb{R}^N} h_1|u_n-u|^{\alpha}dx\Big)^{1/\alpha} \\ &\le (S_0M^{\alpha}\|h_1V^{-t_1}\|_{\infty})^{(\alpha-1)/\alpha} \Big(\int_{\mathbb{R}^N} h_1|u_n-u|^{\alpha}dx\Big)^{1/\alpha}\to 0,\quad \text{as } n\to\infty \end{split} \end{equation} and \begin{equation}\label{2.38} \begin{split} &\int_{\mathbb{R}^N} h_2|u_n|^{\beta-1}|u_n-u|dx \\ &\le \Big(\int_{\mathbb{R}^N} h_2|u_n|^{\beta}dx\Big)^{1-1/\beta} \Big(\int_{\mathbb{R}^N} h_2|u_n-u|^{\beta}dx\Big)^{1/\beta} \\ &\le (S_0M^{\beta}\|h_2V^{-t_2}\|_{\infty})^{1-1/\beta} \Big(\int_{\mathbb{R}^N} h_2|u_n-u|^{\beta}dx\Big)^{1/\beta}\to 0, \quad \text{as } n\to\infty. \end{split} \end{equation} Hence, \begin{equation}\label{2.39} \int_{\mathbb{R}^N} |f(x,u_n)(u_n-u)|dx\le \int_{\mathbb{R}^N} (h_1|u_n|^{\alpha-1}+h_2|u_n|^{\beta-1})|u_n-u|dx \to 0, \end{equation} as $n\to\infty$. This proves the first limit of \eqref{2.31}. The second limit of \eqref{2.31} can be obtained in a similar way. To prove the limit \eqref{2.32}, we use \eqref{1.24} and derive that \begin{equation}\label{2.40} \begin{aligned} &|F(x,u_n)-F(x,v)|\\ &\le C\Big[h_1(x)(|u_n|^{\alpha-1}+|u|^{\alpha-1}) +h_2(x)(|u_n|^{\beta-1}+|u|^{\beta-1})\Big]|u_n-u|. \end{aligned} \end{equation} Then an application of \eqref{2.37} and \eqref{2.38} yields that the limit \eqref{2.32}. The proof is complete. \end{proof} \begin{lemma} \label{lem2.7} Let the assumptions in Theorem \ref{thm1.7} hold. If the sequence $\{u_n\}$ satisfies \eqref{2.29}, then the limits \eqref{2.31} and \eqref{2.32} hold. \end{lemma} \begin{proof} Choose $\psi\in C_0^{\infty}(\mathbb{R})$ such that $supp\psi\subset [-2,2]$ and $\psi(t)=1$ on $[-1,1]$. Denote $g(x,t)=\psi(t)f(x,t)$, $ H(x,t)=(1-\psi(t))f(x,t)$. Then $f(x,t)=g(x,t)+H(x,t)$. Furthermore, from \eqref{1.29}, there exist the constants $a_1,b_1>0$ such that \begin{equation}\label{2.41} |g(x,t)|\le a_1|t|^{\alpha-1},\quad |H(x,t)|\le b_1|t|^{\beta-1},\quad \forall (x,t)\in\mathbb{R}^N\times\mathbb{R}. \end{equation} Denote \begin{equation}\label{2.42} A_n=\int_{\mathbb{R}^N} |g(x,u_n)-g(x,u)|^{\alpha'}dx,\quad D_n=\int_{\mathbb{R}^N} |H(x,u_n)-H(x,u)|^{\beta'}dx, \end{equation} where $t'=t/(t-1)$. By \eqref{2.29}, there exist $\omega_1(x)\in L^{\alpha}(\mathbb{R}^N)$ and $\omega_2(x) \in L^{\beta}(\mathbb{R}^N)$ such that $|u_n(x)|\le \omega_1(x)$ and $|u_n(x)|\le \omega_2(x)$ a.e. in $\mathbb{R}^N$ for all $n\ge 1$. Note that \begin{equation}\label{2.43} \begin{aligned} |g(x,u_n)-g(x,u)|^{\alpha'} & \le C_3(|u_n(x)|^{\alpha}+|u(x)|^{\alpha})\\ &\le C_3(|\omega_1(x)|^{\alpha}+|u(x)|^{\alpha})\equiv d_1(x)\in L^{1}(\mathbb{R}^N) \end{aligned} \end{equation} and \begin{equation}\label{2.44} \begin{aligned} |H(x,u_n)-H(x,u)|^{\beta'} &\le C_3(|u_n(x)|^{\beta}+|u(x)|^{\beta})\\ &\le C_3(|\omega_2(x)|^{\beta}+|u(x)|^{\beta})\equiv d_2(x)\in L^{1}(\mathbb{R}^N), \end{aligned} \end{equation} where $C_3$ is a constant independent of $n$. By the Lebesgue dominated convergence theorem and \eqref{2.29}, we have \begin{equation}\label{2.45} \begin{gathered} \lim_{n\to \infty}A_n = \int_{\mathbb{R}^N} \lim_{n\to \infty}|g(x,u_n) - g(x,u)|^{\alpha'}dx = 0,\\ \lim_{n\to \infty}D_n = \int_{\mathbb{R}^N} \lim_{n\to \infty}|H(x,u_n) - H(x,u)|^{\beta'}dx=0. \end{gathered} \end{equation} Therefore, by the H\"older inequality, \begin{equation}\label{2.46} \begin{split} &\int_{\mathbb{R}^N} |f(x,u_n) - f(x,u)||u_n - u|dx \\ &\le \int_{\mathbb{R}^N} (|g(x,u_n) - g(x,u)|+|H(x,u_n) - H(x,u)|)|u_n - u|dx \\ &\le A_n^{1/\alpha'}\|u_n-u\|_{\alpha}+D_n^{1/\beta'}\|u_n-u\|_{\beta} \\ &\le (A_n^{1/\alpha'}+D_n^{1/\beta'})\|u_n-u\|_E \le M(A_n^{1/\alpha'}+D_n^{1/\beta'}). \end{split} \end{equation} An application of \eqref{2.45} and \eqref{2.46} gives \begin{equation}\label{2.47} \lim_{n\to \infty}\int_{\mathbb{R}^N} f(x,u_n)(u_n-u)dx =\lim_{n\to\infty}\int_{\mathbb{R}^N} f(x,u)(u_n-u)dx. \end{equation} Similarly, from \eqref{2.41}, \eqref{2.43} and \eqref{2.44}, we can derive that \begin{equation}\label{2.48} \lim_{n\to \infty}\int_{\mathbb{R}^N} f(x,u)(u_n-u)dx =\int_{\mathbb{R}^N} \lim_{n\to\infty}f(x,u)(u_n-u)dx=0. \end{equation} Consequently, the limit \eqref{2.31} is given. Similarly, \eqref{2.32} can be proved and the proof is complete. \end{proof} \begin{lemma} \label{lem2.8} Let the assumptions in Theorems \ref{thm1.6} and \ref{thm1.7} hold. Let $\{u_n\}$ be the sequence in Lemmas \ref{lem2.3}--\ref{lem2.5} satisfying \eqref{2.29}. Then $u$ is a critical point of the functional $J$ and $u_n\to u$ in $E$. \end{lemma} \begin{proof} First, we show that $J'(u)=0$ in $E^*$. By Lemmas \ref{lem2.3}--\ref{lem2.5}, the sequence $\{u_n\}$ is bounded in $E$. So, there exists a subsequence, still denoted by $\{u_n\}$, such that $\{u_n\}$ satisfies \eqref{2.29}. Moreover, one sees that for all $\varphi \in C_0^{\infty}(\mathbb{R}^N)$, \begin{equation}\label{2.49} \lim_{n\to\infty}\Big(\int_{\mathbb{R}^N} |\xi|^2(\hat{u_n}-\hat{u })\hat{\varphi} dx +\int_{\mathbb{R}^N} V(x)(u_n-u)\varphi dx\Big)=0. \end{equation} Under assumptions (A3)--(A7), we obtain \begin{equation}\label{2.50} \lim_{n\to\infty}\int_{\mathbb{R}^N} (f(x,u_n)-f(x,u))\varphi dx=0. \end{equation} Furthermore, from \eqref{2.49}, \eqref{2.50} and the assumption $J'(u_n)\to 0$ in $E^*$, we have \begin{equation}\label{2.51} 0=\lim_{n\to\infty}J'(u_n)\varphi=J'(u)\varphi, \quad \forall \varphi \in C_0^{\infty}(\mathbb{R}^N). \end{equation} By the denseness of $C_0^{\infty}(\mathbb{R}^N)$ in $E$, it follows that $J'(u)\varphi=0, \forall\varphi\in E$. Hence, $u$ is a critical point of $J$ in $E$. On the other hand, from \eqref{2.29} it follows that \begin{equation}\label{2.52} R_{n}=\int_{\mathbb{R}^N} |\xi|^{2}\hat{u}(\hat{u}_n-\hat{u})d\xi +\int_{\mathbb{R}^N} V(x)u(u_n-u)dx\to 0,\;\;{\rm as}\quad n\to\infty. \end{equation} Set \begin{equation}\label{2.53} W_{n}:=\int_{\mathbb{R}^N} f(x,u_n)(u_n-u)dx,\quad S_{n}:=\int_{\mathbb{R}^N} f(x,u)(u_n-u)dx,\quad \forall n\in \mathbb{N}. \end{equation} From \eqref{2.31}, it follows that $W_n, S_n\to 0$ as $n\to \infty$ and so \begin{equation}\label{2.54} J'(u)(u_n-u)=R_{n}-S_n\to 0. \end{equation} Similarly, we set \begin{equation}\label{2.55} Q_{n}:=(J'(u_n)-J'(u))(u_n-u)=\|u_n-u\|_E^2-W_n+S_n,\quad \forall n\in \mathbb{N}. \end{equation} Obviously, relation \eqref{2.55} can be reduced to the form \begin{equation}\label{2.56} \|u_n-u\|_E^2=W_{n}+Q_{n}-S_{n},\quad \forall n\in \mathbb{N}. \end{equation} From \eqref{2.53}, \eqref{2.54} and $J'(u_n)\to 0$, we find $Q_{n}\to 0$ and $\|u_n-u\|_E \to 0$ as $n\to \infty$. Thus $u_n\to u$ in $E$ as $n\to\infty$ under assumptions (A3)--(A7). Therefore, $J$ satisfies the $(C)_c$ condition in $E$ and the proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.6}] Clearly, the functional $J$ defined by \eqref{1.22} is even. By Lemma \ref{lem2.8}, the functional $J$ satisfies the $(C)_c$ condition. Next, we prove that $J$ satisfies (A8) and (A9) in Lemma \ref{lem2.1}. From (A3), we have \begin{equation}\label{2.57} J(u)=\frac{1}{2}\|u\|_E^2-\int_{\mathbb{R}^N} F(x,u)dx\ge \frac{1}{2}\|u\|_E^2-\int_{\mathbb{R}^N}(h_1|u|^{\alpha}+h_2|u|^{\beta})dx. \end{equation} Arguing as in the proof of \eqref{2.34} and \eqref{2.35}, we obtain \begin{equation}\label{2.58} J(u)\ge\frac{1}{2}\|u\|_E^2-S_0M_3(\|u\|_E^{\alpha}+\|u\|_E^{\beta}) \ge\rho^2(\frac{1}{2}-2S_0M_3\rho^{\alpha-2})\ge \frac{\rho^2}{4}\equiv \alpha_0>0, \end{equation} where $M_3=\max\{\|h_1V^{-t_1}\|_{\infty}, \|h_2V^{-t_2}\|_{\infty}\}$ and $\|u\|_E=\rho=\min\{1,(8S_0M_3)^{\frac{1}{2-\alpha}}\}$. Thus, by \eqref{2.58}, condition (A8) is satisfied. We now satisfy condition (A9). For any finite dimensional subspace $E_0\subset E$, we assert that there holds $J(u_n)\to -\infty$ when $u_n\in E_0$ and $\|u_n\|_E\to \infty$. Arguing by contradiction, suppose that for some sequence $\{u_n\}\subset E_0$ with $\|u_n\|_E\to \infty$, there is $M_4>0$ such that $J(u_n)\ge -M_4$, for all $n\geq 1$. Set $v_n(x)=\frac{u_n(x)}{\|u_n\|_E}$, then $\|v_n\|_E=1$. Passing to a subsequence, we may assume that $v_n\rightharpoonup v$ in $E$, $v_n(x)\to v(x)$ a.e on $\mathbb{R}^N$. Since $E_0$ is finite dimensional, then $v_n\to v$ in $E_0$ and so $v\not=0$ a.e.in $\mathbb{R}^N$. Set $\Omega=\{x\in\mathbb{R}^N:v(x)\not=0\}$, then meas$(\Omega)>0$. For $x\in\Omega$, we have $\lim_{n\to\infty}|u_n(x)|=\infty$. Then, from (A5) it follows that \begin{equation}\label{2.59} \begin{split} 0&=\limsup_{n\to\infty}\frac{-M}{\|u_n\|_E^{2}} \le \limsup_{n\to\infty}\frac{J(u_n)}{\|u_n\|_E^{2}} = \limsup_{n\to\infty}\Big[\frac{1}{2} -\int_{\mathbb{R}^N} \frac{F(x,u_n)}{|u_n|^{2}}|v_n|^{2}dx\Big] \\ &\le \frac{1}{2}-\int_{\mathbb{R}^N} \liminf_{n\to\infty} \frac{F(x,u_n)}{|u_n|^{2}}\chi_{\Omega}(x)|v_n|^{2}dx =-\infty, \end{split} \end{equation} and we have a contradiction. So, there exists $R=R(E_0)>0$ such that $J(u)<0$ for $u\in E_0$ and $\|u\|_E\ge R$. Therefore, condition (A9) is satisfied. Then an application of Lemma \ref{lem2.1} shows that \eqref{1.1} admits infinitely many solutions $u_n\in E$ with $J(u_n)\to \infty$ as $n\to \infty$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.7}] Clearly, the functional $J$ defined by \eqref{1.22} is even. By Lemma \ref{lem2.8}, the functional $J$ satisfies the $(C)_c$ condition. Next, we prove that $J$ satisfies (A8) and (A9) in Lemma \ref{lem2.1}. From (A7), we have \begin{equation}\label{2.60} J(u)=\frac{1}{2}\|u\|_E^2-\int_{\mathbb{R}^N} F(x,u)dx\ge \frac{1}{2}\|u\|_E^2-C_0\int_{\mathbb{R}^N}(|u|^{\alpha}+|u|^{\beta})dx. \end{equation} Furthermore, from \eqref{1.20} it follows that \begin{equation}\label{2.61} J(u)\ge\frac{1}{2}\|u\|_E^2-C_4(\|u\|_E^{\alpha}+\|u\|_E^{\beta}) \ge\rho^2(\frac{1}{2}-C_4\rho^{\alpha-2})\ge \frac{\rho^2}{4}\equiv \alpha_1>0, \end{equation} with $\|u\|_E=\rho=\min\{1,(4C_4)^{\frac{1}{2-\alpha}}\}$ and $C_4=\max\{S_{\alpha}^{\alpha},S_{\beta}^{\beta}\}$. Thus, by \eqref{2.61}, condition (A8) is satisfied. Similarly, we can derive \eqref{2.59} and the verification of condition (A9) is finished. Again, using Lemma \ref{lem2.1}, we complete the proof of Theorem \ref{thm1.7}. \end{proof} \subsection*{Acknowledgments} This work is supported by the Fundamental Research Funds for the Central Universities of China (2015B31014) and by the National Natural Science Foundation of China (No.11571092). The author would like to express his sincere gratitude to the reviewers for their valuable comments and suggestions. \begin{thebibliography}{99} \bibitem {F1} T. Bartsch, Z. Q. Wang; \emph{Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$}, Commun. Partial Differ. Equ. 20 (1995), 1725-1741. \bibitem {F2} H. Brezis, E. H. Lieb; \emph{A relation between pointwise convergence of functions and convergence of functionals}, Proc. Amer. Math. Soc. 88 (1983), 486-490. \bibitem {F3} X. Cabr\'{e}, J.M. Roquejoffre, Front propagation in Fisher-KPP equations with fractional diffusion. C. R. Acad. Sci. Paris S\'{e}r. I 347(2009)1361-1366. \bibitem {F4} X. Cabr\'{e}, J. Tan; \emph{Positive solutions of nonlinear problems involving the square root of the Laplacian}. Adv. Math. 224 (2010), 2052-2093. \bibitem {F5} L. A. Caffarelli; \emph{Nonlocal equations, drifts and games}, Nonlinear Partial Differential Equations, Abel Symp. 7 (2012), 37-52. \bibitem {F6} X. Chang; \emph{Ground state solutions of asymptotically linear fractional Schr\"odinger equation}, J. Math. Phys. 54 (2013), 061504. \bibitem {F7} 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 {F8} W. Dong, J. F. Xu, Z. L. Wei; \emph{Infinitely many weak solutions for a fractional Schr\"odinger equation}, Boundary Value Problems (2014) 2014: 159. \bibitem {F9} P. Felmer, A. Quaas, J. G. Tan; \emph{Positive solutions of the nonlinear Schr\"odinger equation with the fractional Laplacian}, Proc. Royal Soc. Edinburgh Sect. A 142 (2012), 1237-1262. \bibitem {F10} B. H. Feng; \emph{Ground states for the fractional Schrodinger equation}, Electron. J. Diff. Equ. 2013 (2013) No. 127, 1-11. \bibitem {F11} T. X. Gou, H. R. Sun; \emph{Solutions of nonlinear Schr\"odinger equation with fractional Laplacian without the Ambrosetti-Rabinowitz condition}, Applied Math. Comput. 257 (2015), 409-416. \bibitem {F12} Y. X. Hua, X. H. Yu; \emph{On the ground state solution for a critical fractional Laplacian equation}, Nonlinear Analysis, 87 (2013), 116-125 \bibitem {F13} A. Iannizzotto, M. Squassina; \emph{1/2-Laplacian problems with exponential nonlinearity}, J. Math. Anal. Appl. 414 (2014), 372-385. \bibitem {F15} N. Laskin \emph{Fractional Schr\"odinger equation}, Phys. Rev. 66 (2002), 56-108. \bibitem {F16} N. Laskin; \emph{Fractional quantum mechanics and L\'{e}vy path integrals}, Phys. Lett. A 268 (2000), 298-305. \bibitem {F17} E. Di Nezza, G. Palatucci, E. Valdinoci; \emph{Hitchhiker's guide to the fractional sobolev spaces}, Bull. des Sci. Math. 136 (2012), 521-573. \bibitem {F18} P. Pucci, M. Q. Xiang, B. L. Zhang; \emph{Multiple solutions for nonhomogeneous Schr\"odinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^N$}, Calc. Var. 54 (2015), 2785-2806. \bibitem {F19} P. H. Rabinowitz; \emph{Minimax methods in critical point theory with applications to differential equations}, in: CBMS Reg. Conf. Ser. in Math., vol. 65, Amer. Math. Soc. Providence, RI, (1986) \bibitem {F20} S. Secchi; \emph{Ground state solutions for nonlinear fractional Schr\"odinger equations in $\mathbb{R}^N$}, J. Math. Phys. 54 (2013), 031501. \bibitem {F21} S. Secchi; \emph{On fractional Schr\"odinger equations in $\mathbb{R}^N$ without the Ambrosetti-Rabinowitz condition}, 2014, http://arxiv.org/abs/1210.0755. \bibitem {F22} X. D. Shang, J. H. Zhang; \emph{Ground states for fractional Schr\"odinger equations with critical growth}, Nonlinearity 27 (2014), 187-207. \bibitem {F23} Y. Sire, E. Valdinoci; \emph{Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result}. J. Funct. Analysis 256 (2009), 1842-1864. \bibitem {F24} M. de Souza; \emph{On a class of nonhomogeneous fractional quasilinear equations in $\mathbb{R}^n$ with exponential growth}, Nonlinear Differ. Equ. Appl. 22 (2015), 499-511. \bibitem {F25} M. Struwe; \emph{Variational Methods}, 3rd ed. Springer-Verlag: New York, 2000. \bibitem {F26} X. H. Tang; \emph{Infinitely many solutins for semilinear Schr\"odinger equation with sign-changing potential and nonlinearity}, J. Math. Anal. Appl. 401 (2013), 407-415 \bibitem {F27} K. M. Teng; \emph{Multiple solutions for a class of fractional Schr\"odinger equations in $\mathbb{R}^N$}, Nonlinear Analysis: Real World Applications 21 (2015), 76-86. \bibitem {F28} M. Weinstein; \emph{Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities}, J. Diff. Eqns 69 (1987), 192-203. \bibitem {F29} J. F. Xu, Z. L. Wei, W. Dong; \emph{Existence of weak solutions for a fractional Schr\"odinger equation}, Commun Nonlinear Sci Numer Simulat 22 (2015), 1215-1222. \end{thebibliography} \end{document}