\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 250, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/250\hfil Fourth-order elliptic equations] {Multiplicity and concentration of solutions for fourth-order elliptic equations with mixed nonlinearity} \author[W. Zhang, X. Tang, J. Zhang, Z. Luo \hfil EJDE-2017/250\hfilneg] {Wen Zhang, Xianhua Tang, Jian Zhang, Zhiming Luo} \address{Wen Zhang \newline School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China.\newline Key Laboratory of Hunan Province for Mobile Business Intelligence, Hunan University of Commerce, Changsha, 410205 Hunan, China} \email{zwmathcsu@163.com} \address{Xianhua Tang \newline School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China} \email{tangxh@mail.csu.edu.cn} \address{Jian Zhang \newline School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China.\newline Key Laboratory of Hunan Province for Mobile Business Intelligence, Hunan University of Commerce, Changsha, 410205 Hunan, China} \email{zhangjian433130@163.com} \address{Zhiming Luo (corresponding author) \newline School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China} \email{zhmluo2007@163.com} \dedicatory{Communicated by Paul H. Rabinowtiz} \thanks{Submitted June 16, 2017. Published October 10, 2017.} \subjclass[2010]{35J35, 35J60} \keywords{Fourth-order elliptic equations; concentration; mixed nonlinearity; \hfill\break\indent concave-convex nonlinearity; variational methods} \begin{abstract} This article concerns the fourth-order elliptic equation \begin{gather*} \Delta^2u-\Delta u+\lambda V(x)u=f(x, u)+\mu \xi(x)|u|^{p-2}u, \quad x\in \mathbb{R}^{N},\\ u\in H^2(\mathbb{R}^{N}), \end{gather*} where $\lambda >0$ is a parameter, $V\in C(\mathbb{R}^{N},\mathbb{R})$ and $V^{-1}(0)$ has nonempty interior. Under some mild assumptions, we establish the existence of two nontrivial solutions. Moreover, the concentration of these solutions is explored on the set $V^{-1}(0)$ as $\lambda\to\infty$. As an application, we give the similar results and concentration phenomenona for the above problem with concave and convex nonlinearities. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} This article concerns the fourth-order elliptic equation \begin{equation}\label{1.1} \begin{gathered} \Delta^2u-\Delta u+\lambda V(x)u=f(x, u)+\mu \xi(x)|u|^{p-2}u,\quad x\in\mathbb{R}^N,\\ u\in H^2(\mathbb{R}^N), \end{gathered} \end{equation} where $\Delta^2:=\Delta(\Delta)$ is the biharmonic operator, $V\in C(\mathbb{R}^{N})$, $f\in C(\mathbb{R}^{N}\times \mathbb{R})$, $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$, $\lambda>0$, $\mu>0$ and $10$, where $a$ is a positive constant; \item[(A2)] for any $b>0$,$\operatorname{meas}(V_{b})<+\infty$, where meas denotes the Lebesgue measure and $V_{b}:=\{x\in \mathbb{R}^{N}|V(x)\leq b\}$. \end{itemize} The authors in \cite{YW,YT,ZTZ1,ZTZ2} established the existence of infinitely many solutions under various hypotheses on the nonlinearity. Zhang et al.\ \cite{ZTCZ} studied the sign-changing solutions of problem \eqref{1.1} with Kirchhoff-type. When replacing (A2) by a more general assumption: \begin{itemize} \item[(A3)] there is $b>0$ such that $\operatorname{meas}(V_{b})<+\infty$, \end{itemize} the compactness of the embedding fails and this situation becomes more delicate. Recently, the authors in \cite{LCW,YT1} considered the following equation with a parameter under condition (A3), \begin{gather*} \Delta^2u-\Delta u+\lambda V(x)u=f(x, u), \quad x\in\mathbb{R}^N,\\ u\in H^2(\mathbb{R}^N). \end{gather*} With the aid of a parameter, they proved that the energy functional possess the property of being locally compact. Moreover, the authors of these article proved the existence of infinitely many high energy solutions for superlinear case. For somewhat related sublinear case and the existence of infinitely many small negative-energy solutions, see also \cite{YT,YT1,ZTZ}. For the singularly perturbed problem \begin{equation} \begin{gathered} \epsilon^{4}\Delta^2u+V(x)u=f(u), \quad x\in\mathbb{R}^N,\\ u\in H^2(\mathbb{R}^N), \end{gathered} \end{equation} the authors \cite{PS,PS1} considered when the potential $V$ is positive and has global minimum. They obtained the existence of semi-classical solutions. Moreover, they also shown the concentration phenomenon of semi-classical solutions around global minimum of the potential $V$ as $\epsilon\to0$. Motivated by the above papers, we will consider problem \eqref{1.1} with steep well potential, and study the existence of nontrivial solution and concentration results (as $\lambda\to\infty$). To deduce our statements, we need to make the following assumptions on potential $V$: \vskip2mm \begin{itemize} \item[(A4)] $V(x)\in C(\mathbb{R}^{N},\mathbb{R})$ and $V(x)\geq 0$ on $\mathbb{R}^{N}$; \item[(A5)] $\Omega=\operatorname{int} V^{-1}(0)$ is nonempty and has smooth boundary with $\bar{\Omega}=V^{-1}(0)$. \end{itemize} This kind of hypotheses was first introduced by Bartsch and Wang \cite{BW} (see also \cite{BPW}) in the study of a nonlinear Schr\"{o}dinger equation and the potential $\lambda V(x)$ with $V$ satisfying (A3)--(A5) is referred as the steep well potential. It is worth mentioning that the above papers always assumed the potential $V$ is positive ($V>0$). Compared with the case $V>0$, our assumptions on $V$ are rather weak, and perhaps more important. Generally speaking, there may exist some behaviours and phenomenons for the solutions of problem \eqref{1.1} under condition (A5), such as the concentration phenomenon of solutions. Very recently, in \cite{ZTZ3}, the authors considered this case, and proved the existence and concentration of solutions when the nonlinearity is only sublinear. Besides, we are also interested in the case that the nonlinearity is a more general mixed nonlinearity involving a combination of superlinear ($f(x,u)$) and sublinear ($\xi(x)|u|^{p-2}u$, $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$ and $14$, $2_{\ast}=\infty$ if $N\leq 4$; \item[(A7)] $f(x, u)=o(|u|)$ as $|u|\to 0$ uniformly for $x\in \mathbb{R}^{N}$; \item[(A8)] there exists $\theta >2$ such that $0<\theta F(x, u)\leq uf(x, u)$ for every $x\in \mathbb{R}^{N}$ and $u \neq 0$, where $F(x, u)=\int_0^{u}f(x, t)dt$. \end{itemize} On the existence of solutions we have the following result. \begin{theorem} \label{thm1.1} Assume that the conditions {\rm (A3)--(A8)} hold, and $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$ $(1\Lambda_0$ and $0<\mu<\mu_0$, problem \eqref{1.1} has at least two nontrivial solutions $u_{\lambda}^{i}$ $(i=1, 2)$. \end{theorem} On the concentration of solutions we have the following result. \begin{theorem} \label{thm1.2} Let $u_{\lambda}^{i}$, $(i=1, 2)$ be the solutions of problem \eqref{1.1} obtained in Theorem \ref{thm1.1} and $\mu\in(0,\mu_0)$, then $u_{\lambda}^{i}\to u_0^{i}$ in $H^2(\mathbb{R}^{N})$ as $\lambda\to \infty$, where $u_0^{i}\in H^2(\Omega)\cap H_0^1(\Omega)$ are nontrivial solutions of the equation \begin{equation}\label{1.2} \begin{gathered} \Delta^2u-\Delta u=f(x, u)+\mu \xi(x)|u|^{p-2}u, \quad \text{in }\Omega,\\ u=\Delta u=0,\quad \text{on }\partial\Omega. \end{gathered} \end{equation} \end{theorem} A model of nonlinearity is \begin{equation}\label{1.3} g(x,u):=|u|^{q-2}u+\mu \xi(x)|u|^{p-2}u \end{equation} with $1\Lambda_0$ and $0<\mu<\mu_0$, problem \eqref{1.1} has at least two nontrivial solutions $u_{\lambda}^{i}$ $(i=1, 2)$. \end{corollary} \begin{corollary} \label{coro1.4} Let $u_{\lambda}^{i}$, $(i=1, 2)$ be the solutions of problem \eqref{1.1} obtained in Corollary \ref{coro1.3} and $\mu\in(0,\mu_0)$, then $u_{\lambda}^{i}\to u_0^{i}$ in $H^2(\mathbb{R}^{N})$ as $\lambda\to \infty$, where $u_0^{i}\in H^2(\Omega)\cap H_0^1(\Omega)$ are nontrivial solutions of the equation \begin{equation} \begin{gathered} \Delta^2u-\Delta u=|u|^{q-2}u+\mu \xi(x)|u|^{p-2}u, \quad \text{in }\Omega,\\ u=\Delta u=0,\quad \text{on}~\partial\Omega. \end{gathered} \end{equation} \end{corollary} \begin{remark} \label{rmk1.5} \rm Compared with the previous works, our results seem more general and complete, which is reflected in the following aspects. On the one hand, our assumptions on $V$ are much weaker, and the existence and multiplicity of nontrivial solutions are obtained without any symmetric assumption. On the other hand, more importantly, we also explore the phenomenon of concentrations of these solutions as $\lambda\to\infty$, which seems to be rarely concerned in the previous studies. \end{remark} The rest of this article is organized as follows. In Section $2$, we establish the variational framework associated with problem \eqref{1.1}, and we also give the proof of Theorem \ref{thm1.1}. In Section $3$, we study the concentration of solutions and prove Theorem \ref{thm1.2}. \section{Variational setting and proof of Theorem \ref{thm1.1}} Below by $\|\cdot\|_{s}$ we denote the usual $L^{s}$-norm for $2\leq s\leq 2_{\ast}$, $c_{i}$, $C$, $C_{i}$ stand for different positive constants. Now, we establish the variational setting of problem \eqref{1.1}. Let \[ E=\Big\{u\in H^2(\mathbb{R}^{N}): \int_{\mathbb{R}^N}\left(|\Delta u|^2+|\nabla u|^2+V(x)u^2\right)dx<+\infty\Big\} \] be equipped with the inner product \[ (u,v)=\int_{\mathbb{R}^N}\left(\Delta u \Delta v+\nabla u\cdot \nabla v +V(x)uv\right)dx,\quad u, v\in E, \] and the norm \[ \|u\|=\Big(\int_{\mathbb{R}^N}(|\Delta u|^2+|\nabla u|^2+V(x)u^2)dx\Big)^{1/2},\quad u\in E. \] For $\lambda>0$, we also need the inner product \[ (u,v)_{\lambda}=\int_{\mathbb{R}^N}\left(\Delta u \Delta v+ \nabla u\cdot \nabla v +\lambda V(x)uv\right)dx,\quad u, v\in E, \] and the corresponding norm $\|u\|_{\lambda}^2=(u,u)_{\lambda}$. It is clear that $\|u\|\leq \|u\|_{\lambda}$, for $\lambda \geq 1$. Set $E_{\lambda}=(E, \|\cdot\|_{\lambda})$, then $E_{\lambda}$ is a Hilbert space. By (A3)-(A4) and the statement of proof of \cite[Lemma 2.1]{YT1}, we can demonstrate that there exists a positive constant $\gamma_0$ (independent of $\lambda$) such that \[ \|u\|_{H^2(\mathbb{R}^{N})}\leq \gamma_0\|u\|_{\lambda},\quad \text{for all } u\in E_{\lambda}. \] Furthermore, the embedding $E_{\lambda}\hookrightarrow L^{s}(\mathbb{R}^{N})$ is continuous for $s\in[2,2_{*}]$, and $E_{\lambda}\hookrightarrow L_{\rm loc}^{s}(\mathbb{R}^{N})$ is compact for $s\in[2,2_{*})$, i.e., there are constants $\gamma_{s}, \gamma_0>0$ such that \begin{equation}\label{2.1} \|u\|_{s}\leq \gamma_{s}\|u\|_{{H^2(\mathbb{R}^{N})}} \leq \gamma_{s}\gamma_0\|u\|_{\lambda},\quad \text{for all } u\in E_{\lambda},\; 2\leq s\leq 2_{\ast}. \end{equation} Let \begin{equation}\label{2.2} \Phi_{\lambda}(u)=\frac{1}{2} \int_{\mathbb{R}^N}\left(|\Delta u|^2+|\nabla u|^2+\lambda V(x)u^2\right)dx-\Psi(u), \end{equation} where \[ \Psi(u)=\int_{\mathbb{R}^N}F(x, u)dx+\frac{\mu}{p} \int_{\mathbb{R}^N}\xi(x)|u|^{p}dx. \] By a standard argument and H\"{o}lder inequality, it is easy to verify that $\Phi_{\lambda} \in C^1(E_{\lambda}, \mathbb{R})$ and \begin{equation}\label{2.3} \langle\Phi_{\lambda}'(u), v\rangle=\int_{\mathbb{R}^N}\left[\Delta u \Delta v +\nabla u \cdot \nabla v+\lambda V(x)uv\right]dx-\langle\Psi'(u),v\rangle, \end{equation} for all $u, v\in E_{\lambda}$, where \[ \langle\Psi'(u),v\rangle=\int_{\mathbb{R}^N}f(x, u)vdx+\mu \int_{\mathbb{R}^N}\xi(x)|u|^{p-2}uvdx. \] We say that $I\in C^1(X, \mathbb{R})$ satisfies (PS) condition if any sequence $\{u_{n}\}$ such that $I(u_{n})\to d$, $I'(u_{n})\to 0$ has a convergent subsequence. To prove our result, we need the following Mountain Pass Theorem. \begin{theorem}[{\cite[Theorem 2.2]{Rabinowitz}}] \label{thm2.1} Let $X$ be a real Banach space and $I\in C^1(X, \mathbb{R})$ satisfying (PS) condition. Suppose $I(0)=0$ and \begin{itemize} \item[(1)] there are constants $\rho, \eta >0$ such that $I_{\partial B_{\rho}(0)}\geq \eta$, \item[(2)] there is an constant $e\in X\setminus \bar{B}_{\rho}(0)$ such that $I(e)\leq 0$, then $I$ possesses a critical value $\beta \geq \eta$. \end{itemize} \end{theorem} \begin{lemma} \label{lem2.2} Assume that {\rm (A6), (A7)} are satisfied, and $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$. Then there exist three positive constants $\mu_0$, $\rho$ and $\eta$ such that $\Phi_{\lambda}(u)|_{\|u\|_{\lambda}=\rho}\geq \eta>0$ for all $\mu \in (0, \mu_0)$. \end{lemma} \begin{proof} For any $\varepsilon >0$, it follows from conditions (A6) and (A7) that there exist $C_{\varepsilon}>0$ such that \begin{equation}\label{2.4} F(x, u)\leq \frac{\varepsilon}{2}|u|^2+\frac{C_{\varepsilon}}{q}|u|^{q},\quad \text{for all } u\in E_{\lambda}. \end{equation} Thus, from \eqref{2.1}, \eqref{2.4} and the Sobolev inequality, we have that for all $u\in E_{\lambda}$, \[ \int_{\mathbb{R}^N}F(x,u)dx \leq \frac{\varepsilon}{2}\int_{\mathbb{R}^N}u^2dx+\frac{C_{\varepsilon}}{q} \int_{\mathbb{R}^N}|u|^{q}dx \leq \frac{\gamma_{2}^2\gamma_0^2\varepsilon}{2}\|u\|_{\lambda}^2+ \frac{C_{\varepsilon}\gamma_{q}^{q}\gamma_0^{q}}{q}\|u\|_{\lambda}^{q}, \] which implies that \begin{equation}\label{2.5} \begin{aligned} \Phi_{\lambda}(u) &=\frac{1}{2}\|u\|_{\lambda}^2-\int_{\mathbb{R}^N}F(x,u)dx- \frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)|u|^{p}dx\\ &\geq \frac{1}{2}\|u\|_{\lambda}^2-\frac{\gamma_{2}^2\gamma_0^2\varepsilon}{2}\|u\|_{\lambda}^2 -\frac{C_{\varepsilon}\gamma_{q}^{q}\gamma_0^{q}}{q}\|u\|_{\lambda}^{q} -\frac{\mu \gamma_{2}^{p}\gamma_0^{p}}{p}\|\xi\|_{\frac{2}{2-p}}\|u\|_{\lambda}^{p}\\ &=\|u\|_{\lambda}^{p}\Big[\frac{1}{2}\left(1-\gamma_{2}^2\gamma_0^2\varepsilon\right) \|u\|_{\lambda}^{2-p} -\frac{C_{\varepsilon}\gamma_{q}^{q}\gamma_0^{q}}{q}\|u\|_{\lambda}^{q-p} -\frac{\mu \gamma_{2}^{p}\gamma_0^{p}}{p}\|\xi\|_{\frac{2}{2-p}}\Big]. \end{aligned} \end{equation} Take $\varepsilon =\frac{1}{2\gamma_{2}^2\gamma_0^2}$ and define \[ g(t)=\frac{1}{4}t^{2-p}-\frac{C_{\varepsilon}\gamma_{q}^{q}\gamma_0^{q}}{q}t^{q-p}, \quad \text{for }t\geq 0. \] It is easy to prove that there exists $\rho >0$ such that \[ \max_{t\geq 0}g(t)=g(\rho)=\frac{q-2}{4(q-p)}\left[\frac{(2-p)q} {4C_{\varepsilon}\gamma_{q}^{q}\gamma_0^{q}(q-p)}\right]^{\frac{2-p}{q-2}}. \] Then it follows from \eqref{2.5} that there exist positive constants $\mu_0$ and $\eta$ such that $\Phi_{\lambda}(u)|_{\|u\|_{\lambda}=\rho}\geq \eta$ for all $\mu \in (0, \mu_0)$. \end{proof} \begin{lemma} \label{lem2.3} Assume that {\rm (A6)--(A8)} are satisfied, and $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$. Let $\rho$ be as in Lemma \ref{lem2.2}. Then there exists $e\in E_{\lambda}$ with $\|e\|_{\lambda}>\rho$ such that $\Phi_{\lambda}(e)<0$ for all $\mu\geq0$. \end{lemma} \begin{proof} By \eqref{2.4} and (A8), there exists $c>0$ such that \[ F(x, u)\geq c\left(|u|^{\theta}-|u|^2\right),\quad \forall (x, u)\in \mathbb{R}^N\times \mathbb{R}. \] Thus, for $t> 0$, $u\in E_{\lambda}$, we have \begin{align*} \Phi_{\lambda}(tu) &=\frac{t^2}{2} \|u\|_{\lambda}^2 -\int_{\mathbb{R}^N}F(x, tu)dx - \frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)|tu|^{p}dx\\ &\leq\frac{t^2}{2} \|u\|_{\lambda}^2-ct^{\theta}\int_{\mathbb{R}^N}|u|^{\theta}dx +ct^2 \int_{\mathbb{R}^N}|u|^2dx- \frac{\mu}{p}t^{p}\int_{\mathbb{R}^N}\xi(x)|u|^{p}dx, \end{align*} which implies that $\Phi_{\lambda}(tu)\to-\infty$ as $t\to\infty$. Therefore, there exist $t_0>0$ and $e:=t_0u$ with $\|e\|_{\lambda}>\rho$ such that $\Phi_{\lambda}(e)<0$. This completes the proof. \end{proof} To find the critical points of $\Phi_{\lambda}$, we shall show that $\Phi_{\lambda}$ satisfies the (PS) condition, i.e. any (PS) sequence $\{u_{n}\}$ has a convergent subsequence in $E_{\lambda}$. Since there is no compactness of the Sobolev embedding, the situation is more difficult. To overcome this difficulty, we need the following convergence results. \begin{lemma} \label{lem2.4} Suppose that $u_{n}\rightharpoonup u_0$ in $E_{\lambda}$. Then, passing to a subsequence \begin{gather}\label{2.6} \Phi_{\lambda}(u_{n})=\Phi_{\lambda}(u_{n}-u_0)+\Phi_{\lambda}(u_0)+o(1), \\ \label{2.7} \Phi'_{\lambda}(u_{n})=\Phi'_{\lambda}(u_{n}-u_0)+\Phi'_{\lambda}(u_0)+o(1) \quad \text{as } n\to \infty. \end{gather} Particularly, if $\{u_{n}\}$ is a (PS) sequence such that $\Phi_{\lambda}(u_{n})\to d$ for some $d\in \mathbb{R}$, then \begin{equation}\label{2.8} \Phi_{\lambda}(u_{n}-u_0)\to d-\Phi_{\lambda}(u_0)\quad \text{and}\quad \Phi'_{\lambda}(u_{n}-u_0)\to0 \end{equation} after passing to a subsequence. \end{lemma} \begin{proof} Since $u_{n}\rightharpoonup u_0$ in $E_{\lambda}$, we have \[ (u_{n}, u_0)_{\lambda}\to (u_0, u_0)_{\lambda},\quad \text{as } n\to \infty. \] which yields \begin{align*} \|u_{n}\|_{\lambda}^2&=(u_{n}-u_0, u_{n}-u_0)_{\lambda}+(u_0, u_{n})_{\lambda}+(u_{n}-u_0, u_0)_{\lambda}\\ &=\|u_{n}-u_0\|_{\lambda}^2+\|u_0\|_{\lambda}^2+o(1). \end{align*} It is clear that \[ (u_{n}, \phi)_{\lambda}=(u_{n}-u_0, \phi)_{\lambda}+(u_0, \phi)_{\lambda}\quad \text{for all }\phi\in E_{\lambda}. \] Hence, to obtain \eqref{2.6} and \eqref{2.7}, it sufficient to check that \begin{gather}\label{2.9} \int_{\mathbb{R}^N} \left[F(x, u_{n})-F(x, u_{n}-u_0)-F(x, u_0)\right]dx=o(1), \\ \label{2.10} \int_{\mathbb{R}^N}\xi(x) \left[|u_{n}|^{p}-|u_{n}-u_0|^{p}-|u_0|^{p}\right]dx=o(1), \\ \label{2.11} \int_{\mathbb{R}^N}\left(f(x, u_{n})-f(x, u_{n}-u_0)-f(x, u_0)\right)\phi dx =o(1)\quad\forall \phi\in E_{\lambda}, \\ \int_{\mathbb{R}^N}\xi(x)\left(|u_{n}|^{p-2}u_{n}-|u_{n}-u_0|^{p-2}(u_{n}-u_0) -|u_0|^{p-2}u_0\right)\phi dx =o(1) \nonumber\\ \text{for all }\phi\in E_{\lambda}. \label{2.12} \end{gather} Here, we only prove \eqref{2.9}$ and \eqref{2.10}$, the verification of \eqref{2.11} and \eqref{2.12} is similar. Take $\omega_{n}:=u_{n}-u_0$, we have $\omega_{n}\rightharpoonup 0$ in $E_{\lambda}$ and $\omega_{n}(x)\to 0$ a.e. $x\in \mathbb{R}^{N}$. It follows from (A6) and (A7) that \begin{gather}\label{2.13} |f(x, u)|\leq \varepsilon |u|+C_{\varepsilon}|u|^{q-1}\quad \forall (x, u)\in \mathbb{R}^N\times \mathbb{R}, \\ \label{2.14} |F(x, u)|\leq \int_0^1|f(x, tu)||u|dt\leq \varepsilon |u|^2 +C_{\varepsilon}|u|^{q},~~\forall (x, u)\in \mathbb{R}^N\times \mathbb{R}. \end{gather} Then \begin{align*} |F(x, \omega_{n}+u_0)-F(x, \omega_{n})| &\leq \int_0^1|f(x, \omega_{n}+\zeta u_0)||u_0|d\zeta\\ &\leq \int_0^1\left(\varepsilon |\omega_{n}+\zeta u_0||u_0| +C_{\varepsilon}|\omega_{n}+\zeta u_0|^{q-1}|u_0|\right)d\zeta\\ &\leq c_{1}\left(\varepsilon |\omega_{n}||u_0|+\varepsilon |u_0|^2 +C_{\varepsilon}|\omega_{n}|^{q-1}|u_0| +C_{\varepsilon}|u_0|^{q}\right). \end{align*} By Young's inequality, we have \[ |F(x, \omega_{n}+u_0)-F(x, \omega_{n})|\leq c_{2}\left(\varepsilon |\omega_{n}|^2 +\varepsilon |u_0|^2+\varepsilon|\omega_{n}|^{q} +C_{\varepsilon}|u_0|^{q}\right), \] so that, using \eqref{2.14}, we obtain \[ |F(x, \omega_{n}+u_0)-F(x, \omega_{n})-F(x, u_0)| \leq c_{3}\left(\varepsilon |\omega_{n}|^2 +\varepsilon |u_0|^2+\varepsilon|\omega_{n}|^{q} +C_{\varepsilon}|u_0|^{q}\right), \] for $n\in \mathbb{N}$. Let \[ H_{n}(x):=\max\left\{\left|F(x, \omega_{n}+u_0)-F(x, \omega_{n})-F(x, u_0)\right| -c_{3}\varepsilon \left(|\omega_{n}|^2+|\omega_{n}|^{q}\right), 0\right\}. \] It follows that \[ 0\leq H_{n}(x)\leq c_{3}\left(\varepsilon |u_0|^2 +C_{\varepsilon}|u_0|^{q}\right)\in L^1(\mathbb{R}^N). \] Thus, using Lebesgue dominated convergence theorem, \begin{equation}\label{2.15} \int_{\mathbb{R}^N}H_{n}(x)dx\to 0, \quad \text{as } n\to \infty. \end{equation} From the definition of $H_{n}(x)$, we have \[ |F(x, \omega_{n}+u_0)-F(x, \omega_{n})-F(x, u_0)| \leq c_{3} \varepsilon \left(|\omega_{n}|^2 + |\omega_{n}|^{q}\right)+H_{n}(x), \] for all $n\in \mathbb{N}$. which, together with \eqref{2.15} and \eqref{2.1}, we obtain \[ \int_{\mathbb{R}^N}|F(x, \omega_{n}+u_0)-F(x, \omega_{n}) -F(x, u_0)|dx \leq c_{3} \varepsilon \left(\|\omega_{n}\|_{2}^2 +\|\omega_{n}\|_{q}^{q}\right)+\varepsilon \leq c_{4}\varepsilon, \] for $n$ sufficiently large, hence \[ \int_{\mathbb{R}^N} \left[F(x, u_{n})-F(x, u_{n}-u_0)-F(x, u_0)\right]dx=o(1) \] that is, \eqref{2.9} holds. Observe that $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$, thus, for any $\epsilon>0$ we can choose $R_{\epsilon}>0$ such that \begin{equation}\label{2.16} \Big(\int_{\mathbb{R}^N\setminus B_{R_{\epsilon}}}|\xi(x)|^{\frac{2}{2-p}}dx\Big) ^{\frac{2-p}{2}}<\epsilon. \end{equation} By Sobolev's embedding theorem, $u_{n} \rightharpoonup u_0$ in $E_{\lambda}$ implies $u_{n}\to u_0$ in $L_{\rm loc}^2( \mathbb{R}^{N})$, and hence, \begin{equation}\label{2.17} \lim_{n\to \infty}\int_{B_{R_{\epsilon}}} |u_{n}-u_0|^2dx=0. \end{equation} By \eqref{2.17}, there exists $N_0\in \mathbb{N}$ such that \begin{equation}\label{2.18} \int_{B_{R_{\epsilon}}} |u_{n}-u_0|^2dx<\epsilon ^2,\quad \text{for}~n\geq N_0. \end{equation} Hence, by \eqref{2.1}, \eqref{2.18} and the H\"{o}lder inequality, for any $n\geq N_0$, we have \begin{equation}\label{2.19} \begin{aligned} &\frac{\mu}{p}\int_{B_{R_{\epsilon}}}\xi(x)|u_{n}-u_0|^{p}dx \\ &\leq \frac{\mu}{p} \Big(\int_{B_{R_{\epsilon}}}|\xi(x)|^{\frac{2}{2-p}}dx\Big)^{\frac{2-p}{2}} \Big(\int_{B_{R_{\epsilon}}} |u_{n}-u_0|^2dx\Big)^{p/2}\\ &\leq \frac{\mu}{p}\epsilon^{p}\|\xi(x)\|_{\frac{2}{2-p}}. \end{aligned} \end{equation} On the other hand, by \eqref{2.1} and \eqref{2.16}, we have \begin{equation}\label{2.20} \begin{aligned} &\frac{\mu}{p}\int_{\mathbb{R}^N\setminus B_{R_{\epsilon}}}\xi(x)|u_{n}-u_0|^{p}dx \\ &\leq \frac{\mu}{p}\Big(\int_{\mathbb{R}^N\setminus B_{R_{\epsilon}}} |\xi(x)|^{\frac{2}{2-p}}dx\Big)^{\frac{2-p}{2}} \Big(\int_{\mathbb{R}^N\setminus B_{R_{\epsilon}}} |u_{n}-u_0|^2dx\Big)^{p/2}\\ &\leq \frac{\mu}{p}\epsilon \left(\|u_{n}\|_{2}^{p}+\|u_0\|_{2}^{p}\right)\\ &\leq \frac{\mu}{p}\epsilon \gamma_{2}^{p}\gamma_0^{p} \left(\|u_{n}\|_{\lambda}^{p}+\|u_0\|_{\lambda}^{p}\right)\\ &\leq \frac{\mu}{p}\epsilon \gamma_{2}^{p}\gamma_0^{p} \left(c_{5}^{p}+\|u_0\|_{\lambda}^{p}\right). \end{aligned} \end{equation} Since $\epsilon$ is arbitrary, combining \eqref{2.19} with \eqref{2.20}, we have \begin{gather}\label{2.21} \frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)|u_{n}-u_0|^{p}dx=o(1), \\ \frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)\left(|u_{n}|^{p}-|u_0|^{p}\right)dx \leq \frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)|u_{n}-u_0|^{p}dx. \nonumber \end{gather} Therefore \[ \frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)\left(|u_{n}|^{p}- |u_{n}-u_0|^{p}-|u_0|^{p}\right)=o(1), \] that is, \eqref{2.10} holds. Now, we consider the case $\{u_{n}\}$ is a (PS) sequence such that $\Phi_{\lambda}(u_{n})\to d$ and $\Phi_{\lambda}'(u_{n})\to 0$. It follows from \eqref{2.6} and \eqref{2.7} that \begin{equation}\label{2.22} \Phi_{\lambda}(u_{n}-u_0)=d-\Phi_{\lambda}(u_0)+o(1), \quad \Phi'_{\lambda}(u_{n}-u_0)=-\Phi'_{\lambda}(u_0)+o(1), \end{equation} we show that $\Phi'_{\lambda}(u_0)=0$. For every $\psi \in C_0^{\infty}(\mathbb{R}^N)$, it follows from \eqref{2.13} and the fact that $u_{n}\to u_0$ in $L_{\rm loc}^{s}(\mathbb{R}^N)$ that \[ \int_{\mathbb{R}^N}\left(f(x, u_{n})-f(x, u_0)\right)\psi dx =\int_{\operatorname{supp}\psi}\left(f(x, u_{n})-f(x, u_0)\right)\psi dx=o(1) \] and \begin{align*} &\mu\int_{\mathbb{R}^N}\xi(x)\left(|u_{n}|^{p-2}u_{n}-|u_0|^{p-2}u_0\right) \psi dx \\ &=\mu\int_{\operatorname{supp}\psi}\xi(x)\left(|u_{n}|^{p-2}u_{n} -|u_0|^{p-2}u_0\right)\psi dx=o(1) \end{align*} which implies \[ \langle \Phi'_{\lambda}(u_0), \psi \rangle =\lim_{n\to \infty}\langle \Phi'_{\lambda}(u_{n}), \psi \rangle=0. \] Hence, $\Phi'_{\lambda}(u_0)=0$, which together with the second equation of \eqref{2.22} shows that $\Phi'_{\lambda}(u_{n}-u_0)\to 0$ as $n\to \infty$. Consequently, \eqref{2.8} holds and the proof is complete. \end{proof} \begin{lemma} \label{lem2.5} Let {\rm (A3)--(A5), (A6)--(A8)} be satisfied, there exists $\Lambda_0>0$, any (PS) sequence of $\Phi_{\lambda}$ has a convergent subsequence for all $\lambda\geq\Lambda_0$. \end{lemma} \begin{proof} We adapt an argument in \cite{DS}. Let $\{u_{n}\}$ be a sequence such that $\Phi_{\lambda}(u_{n})\to d$ and $\Phi'_{\lambda}(u_{n})\to 0$ for some $d\in \mathbb{R}$; thus \begin{align*} 1+d+\|u_{n}\|_{\lambda} &\geq \Phi_{\lambda}(u_{n})- \frac{1}{\theta}\langle \Phi'_{\lambda}(u_{n}), u_{n} \rangle\\ &=(\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\lambda}^2 +\int_{\mathbb{R}^N}\big[\frac{1}{\theta}u_{n}f(x, u_{n})-F(x, u_{n})\big]dx \\ &\quad +\int_{\mathbb{R}^N}(\frac{1}{\theta}-\frac{1}{p})\mu\xi(x)|u_{n}|^{p}dx, \end{align*} hence \begin{align*} &1+d+\|u_{n}\|_{\lambda}+(\frac{1}{p}- \frac{1}{\theta})\mu\int_{\mathbb{R}^N}\xi(x)|u_{n}|^{p}dx \\ &\geq (\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\lambda}^2 +\int_{\mathbb{R}^N}\left[\frac{1}{\theta}u_{n}f(x, u_{n})-F(x, u_{n})\right]dx. \end{align*} Since \begin{align*} (\frac{1}{p}-\frac{1}{\theta})\mu\int_{\mathbb{R}^N}\xi(x)|u_{n}|^{p}dx &\leq (\frac{1}{p}-\frac{1}{\theta})\mu \Big(\int_{\mathbb{R}^N}|\xi(x)|^{\frac{2}{2-p}}dx\Big)^{\frac{2-p}{2}} \Big(\int_{\mathbb{R}^N}|u_{n}|^2dx\Big)^{p/2}\\ &=(\frac{1}{p}-\frac{1}{\theta})\mu \|\xi\|_{\frac{2}{2-p}}\|u_{n}\|_{2}^{p}\\ &\leq (\frac{1}{p}-\frac{1}{\theta})\mu\gamma_{2}^{p}\gamma_0^{p} \|\xi\|_{\frac{2}{2-p}}\|u_{n}\|_{\lambda}^{p}. \end{align*} Hence, \begin{align*} &1+d+\|u_{n}\|_{\lambda}+(\frac{1}{p}- \frac{1}{\theta})\mu\gamma_{2}^{p}\gamma_0^{p}\|\xi\|_{\frac{2}{2-p}} \|u_{n}\|_{\lambda}^{p}\\ &\geq (\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\lambda}^2 +\int_{\mathbb{R}^N}\big[\frac{1}{\theta}u_{n}f(x, u_{n})-F(x, u_{n})\big]dx\\ &\geq (\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\lambda}^2. \end{align*} This proves that $\{u_{n}\}$ is bounded in $E_{\lambda}$. Then, passing to a subsequence, we may assume that $u_{n}\rightharpoonup u_0$ in $E_{\lambda}$. Taking $\omega_{n}:=u_{n}-u_0$, we have \begin{equation}\label{2.23} \begin{aligned} \|\omega_{n}\|_{2}^2&\leq \frac{1}{\lambda b} \int_{\{x\in\mathbb{R}^{N}:V(x)>b\}}\lambda V(x)\omega_{n}^2dx +\int_{V_{b}}\omega_{n}^2dx\\ &\leq \frac{1}{\lambda b}\|\omega_{n}\|_{\lambda}^2+o(1), \end{aligned} \end{equation} since $\omega_{n}\rightharpoonup 0$ in $E_{\lambda}$ and $V(x)0$ such that \begin{equation}\label{2.26} \big|\int_{\mathbb{R}^N}\mathcal{F}(x, \omega_{n})dx\big|\leq M. \end{equation} Now we note that $\frac{q}{q-2}> \max\{1, \frac{N}{4}\}$ because $q\in(2, 2_{\ast})$. Fix $\tau \in \big( \max\{1, \frac{N}{4}\}, \frac{q}{q-2}\big)$, from \eqref{2.13}, we know if $|u|\geq 1$, then $|f(x, u)|\leq c_{6}|u|^{q-1}$. Choose $R_{1}$ so large that $\frac{1}{\theta}\leq\frac{1}{2}-\frac{c_{6}^{\tau-1}}{|u|^{q-(q-2)\tau}}$, whenever $|u|\geq R_{1}$. Then, for $|u|$ large enough, we have \begin{align*} 0\leq F(x, u)\leq \frac{1}{\theta}uf(x, u) &\leq \big[\frac{1}{2}-\frac{c_{6}^{\tau-1}}{|u|^{q-(q-2)\tau}}\big]uf(x, u)\\ &\leq \big[\frac{1}{2}-\frac{|f(x, u)| ^{\tau-1}}{|u|^{\tau+1}}\big]uf(x, u), \end{align*} which implies that, for $|u|$ sufficiently large \begin{equation}\label{2.27} \frac{\left|f(x, u)\right| ^{\tau}}{|u|^{\tau}} \leq \frac{1}{2}uf(x, u)-F(x, u)=\mathcal{F}(x, u). \end{equation} Combining this with \eqref{2.24}$, \eqref{2.26}$ with $\sigma=\frac{2\tau}{\tau-1}\in(2, 2_{\ast})$ and the H\"{o}lder inequality, we obtain for large $n$, \begin{equation}\label{2.28} \begin{aligned} &\int_{|\omega_{n}|\geq R_{1}}f(x, \omega_{n})\omega_{n}dx\\ &\leq\Big(\int_{|\omega_{n}|\geq R_{1}}\big| \frac{f(x, \omega_{n})}{\omega_{n}}\big|^{\tau} dx\Big)^{1\tau}\Big(\int_{|\omega_{n}|\geq R_{1}}|\omega_{n}|^{\sigma} dx\Big)^{2/\sigma}\\ &\leq \Big(\int_{|\omega_{n}|\geq R_{1}}\mathcal{F}(x, \omega_{n})dx\Big)^{1\tau} \|\omega_{n}\|_{\sigma}^2\\ &\leq M^{1\tau}(\gamma_{q}\gamma_0)^{\frac{2q(\sigma-2)}{(q-2)\sigma}} \big(\frac{1}{\lambda b}\big)^{\frac{2(q-\sigma)}{(q-2)\sigma}} \|\omega_{n}\|_{\lambda}^2+o(1)\\ &=c_{7}(\frac{1}{\lambda b})^{\theta_{1}}\|\omega_{n}\|_{\lambda}^2+o(1). \end{aligned} \end{equation} where $c_{7}=M^{1\tau}(\gamma_{q}\gamma_0)^{\frac{2q(\sigma-2)}{(q-2)\sigma}} >0$, $\theta_{1}=\frac{2(q-s)}{s(q-2)}>0$. In addition, using \eqref{2.13} and \eqref{2.24}, we have \begin{equation}\label{2.29} \begin{aligned} \int_{|\omega_{n}|\leq R_{1}}f(x, \omega_{n})\omega_{n}dx &\leq \int_{|\omega_{n}|\leq R_{1}}\left(\epsilon+C_{\epsilon}R_{1}^{q-2}\right) \omega_{n}^2dx\\ &\leq \frac{C_{\epsilon}R_{1}^{q-2}}{\lambda b}\|\omega_{n}\|_{\lambda}^2+o(1)\\ &=\frac{c_{8}}{\lambda b}\|\omega_{n}\|_{\lambda}^2+o(1), \end{aligned} \end{equation} where $c_{8}=C_{\epsilon}R_{1}^{q-2}$. Consequently, combining \eqref{2.21}, \eqref{2.28} with \eqref{2.29}, we obtain \begin{align*} o(1)&=\langle\Phi'_{\lambda}(\omega_{n}), \omega_{n}\rangle\\ &=\|\omega_{n}\|_{\lambda}^2-\int_{\mathbb{R}^N}f(x, \omega_{n})\omega_{n}dx -\mu \int_{\mathbb{R}^N}\xi(x)|\omega_{n}|^{p}dx\\ &\geq \big[1-\frac{c_{8}}{\lambda b}-c_{7}\big(\frac{1}{\lambda b}\big)^{\theta_{1}} \big]\|\omega_{n}\|_{\lambda}^2+o(1). \end{align*} Choosing $\Lambda_0>0$ large enough such that the term in the brackets above is positive when $\lambda> \Lambda_0$, we obtain $\omega_{n}\to 0$ in $E_{\lambda}$, thus $u_{n}\to u_0$ in $E_{\lambda}$. This completes the proof. \end{proof} Define \[ d_{\lambda}=\inf_{\gamma\in\Gamma_{\lambda}}\max_{0\leq t\leq1} \Phi_{\lambda}(\gamma(t)) \] where $\Gamma_{\lambda}=\big\{\gamma\in C([0,1],E_{\lambda}) :\gamma(0)=0,\gamma(1)=e\big\}$. \begin{proof}[Proof of Theorem \ref{thm1.1}] By Theorem \ref{thm2.1}, and Lemmas \ref{lem2.2} and \ref{lem2.3}, we obtain that, for each $\lambda\geq \Lambda_0$, $0<\mu<\mu_0$, there exists (PS) sequence $\{u_{n}\}\subset E_{\lambda}$ for $\Phi_{\lambda}$ on $E_{\lambda}$. Then, by Lemma \ref{lem2.5}, we can conclude that there exist a subsequence $\{u_{n}\}\subset E_{\lambda}$ and $u_{\lambda}^1\in E_{\lambda}$ such that $u_{n}\to u_{\lambda}^1$ in $E_{\lambda}$. Moreover, $\Phi_{\lambda}(u_{\lambda}^1)=d_{\lambda}\geq \eta >0$. The second solution of problem \eqref{1.1} will be constructed through the local minimization. Since $\xi\in L^{\frac{2}{2-p}}(\mathbb{R}^{N},\mathbb{R}^{+})$, we can choose a function $\phi \in E_{\lambda}$ such that \[ \int_{\mathbb{R}^N}\xi(x)|\phi|^{p}dx>0. \] Thus, by (A8) we have \begin{equation}\label{2.30} \begin{aligned} \Phi_{\lambda}(l\phi) &=\frac{l^2}{2} \|\phi\|_{\lambda}^2 -\int_{\mathbb{R}^N}F(x, l\phi)dx -\frac{\mu l^{p}}{p}\int_{\mathbb{R}^N}\xi(x)|\phi|^{p}dx\\ &\leq\frac{l^2}{2} \|\phi\|_{\lambda}^2- \frac{\mu l^{p}}{p}\int_{\mathbb{R}^N}\xi(x)|\phi|^{p}dx <0, \end{aligned} \end{equation} for $l>0$ small enough. Hence, there exists $\rho_{1}>0$ such that $\beta:=\inf\{\Phi_{\lambda}(u): u\in \bar{B}_{\rho_{1}}\}<0$. By the Ekeland's variational principle, there exists a minimizing sequence $\{u_{n}\}\subset \bar{B}_{\rho_{1}}$ such that $\Phi_{\lambda}(u_{n})\to \beta$ and $\Phi'_{\lambda}(u_{n})\to 0$ as $n\to \infty$. Hence, Lemma \ref{lem2.5} implies that there exists a nontrivial solution $u_{\lambda}^2$ of problem \eqref{1.1} satisfying \[ \Phi_{\lambda}(u_{\lambda}^2)<0\quad \text{and}\quad \|u_{\lambda}^2\|_{\lambda}<\rho_{1}. \] Moreover, \eqref{2.30} implies that there exists $l_0>0$ and $\kappa <0$ are independent of $\lambda$ such that $\Phi_{\lambda}(l_0\phi)=\kappa$ and $\|l_0\phi\|_{\lambda}<\rho_{1}$. Therefore, we can conclude that \[ \Phi_{\lambda}(u_{\lambda}^2)\leq \kappa <0< \eta < d_{\lambda}= \Phi_{\lambda}(u_{\lambda}^1)\quad \text{for all } \lambda> \Lambda_0\text{ and } 0<\mu<\mu_0. \] This completes the proof. \end{proof} \section{Concentration of solutions} Here we study the concentration of solutions and give the proof of Theorem \ref{thm1.2}. Define \[ d_0=\inf_{\gamma\in\widetilde{\Gamma}_{\lambda}} \max_{0\leq t\leq1}\Phi_{\lambda}|_{H^2(\Omega)\cap H_0^1(\Omega)}(\gamma(t)) \] where \[ \widetilde{\Gamma}_{\lambda}=\big\{\gamma\in C([0,1], H^2(\Omega)\cap H_0^1(\Omega)):\gamma(0)=0,\gamma(1)=e\big\}, \] and $\Phi_{\lambda}|_{H^2(\Omega)\cap H_0^1(\Omega)}$ is a restriction of $\Phi_{\lambda}$ on $H^2(\Omega)\cap H_0^1(\Omega)$. Note that \[ \Phi_{\lambda}|_{H^2(\Omega)\cap H_0^1(\Omega)}(u) =\frac{1}{2}\int_{\Omega}(|\Delta u|^2+|\nabla u|^2)dx -\int_{\Omega}F(x,u)dx-\mu\int_{\Omega}\xi(x)|u|^{p}dx \] and $d_0$ independent of $\lambda$. From the above arguments, we conclude that functional $\Phi_{\lambda}|_{H^2(\Omega)\cap H_0^1(\Omega)}$ has a mountain pass type solution $\tilde{u}$ such that $\Phi_{\lambda}|_{H^2(\Omega)\cap H_0^1(\Omega)}(\tilde{u})=d_0$. Since $(H^2(\Omega)\cap H_0^1(\Omega))\subset E_{\lambda}$ for all $\lambda>0$, it is easy to see that $0<\eta< d_{\lambda}d_0$, thus \[ 0<\eta< d_{\lambda}0, R_0>0$ and $x_{n}\in \mathbb{R}^{N}$ such that \[ \int_{B_{R_0}(x_{n})}|u_{n}^{(i)}-u_0^{i}|^2dx\geq \delta. \] Since $u_{n}^{i}\to u_0^{i}$ in $L_{\rm loc}^2(\mathbb{R}^{N})$, $|x_{n}|\to \infty$. Hence $\operatorname{meas}\left(B_{R_0}(x_{n})\cap V_{b}\right)\to 0$. By H\"{o}lder's inequality, we have \begin{align*} &\int_{B_{R_0}(x_{n})\cap V_{b}}|u_{n}^{i}-u_0^{i}|^2dx\\ &\leq \left(\operatorname{meas}\left(B_{R_0}(x_{n})\cap V_{b}\right)\right) ^{\frac{2_{*}-2}{2_{*}}} \Big(\int_{\mathbb{R}^{N}}|u_{n}^{i}-u_0^{i}|^{2_{*}}\Big)^{2/2_*} \to 0. \end{align*} Consequently, \begin{align*} \|u_{n}^{i}\|_{\lambda_{n}}^2&\geq \lambda_{n}b \int_{B_{R_0}(x_{n})\cap \{x\in \mathbb{R}^{N}: V(x)\geq b\}}|u_{n}^{i}|^2dx\\ &=\lambda_{n}b\int_{B_{R_0}(x_{n})\cap \{x\in \mathbb{R}^{N}: V(x)\geq b\}}|u_{n}^{i}-u_0^{i}|^2dx\\ &=\lambda_{n}b\Big(\int_{B_{R_0}(x_{n})}|u_{n}^{i}-u_0^{i}|^2dx -\int_{B_{R_0}(x_{n})\cap V_{b}}|u_{n}^{i}-u_0^{i}|^2dx+o(1)\Big)\\ &\to \infty, \end{align*} which contradicts \eqref{3.2}. Next, we show that $u_{n}^{i}\to u_0^{i}$ in $H^2(\mathbb{R}^{N})$. From $\langle\Phi'_{\lambda_{n}}(u_{n}^{i}), u_{n}^{i}\rangle=\langle \Phi'_{\lambda_{n}}(u_{n}^{i}), u_0^{i}\rangle=0$ and the fact that $u_{n}^{i}\to u_0^{i}$ in $L^{q}(\mathbb{R}^{N})$ for $2\leq q< 2_{\ast}$, we have \[ \lim_{n\to \infty}\|u_{n}^{i}\|_{\lambda_{n}}^2=\lim_{n\to \infty}(u_{n}^{i}, u_0^{i})_{\lambda_{n}}=\lim_{n\to \infty}(u_{n}^{i}, u_0^{i})=\|u_0^{i}\|^2, \] therefore \[ \limsup\limits_{n\to\infty}\|u_{n}^{i}\|^2\leq \|u_0^{i}\|^2. \] On the other hand, the weak lower semi-continuity of norm yields \[ \|u_0^{i}\|^2\leq \liminf_{n\to \infty} \|u_{n}^{i}\|^2\leq \limsup_{n\to \infty} \|u_{n}^{i}\|^2\leq \lim_{n\to \infty} \|u_{n}^{i}\|_{\lambda_{n}}^2, \] thus, $u_{n}^{i}\to u_0^{i}$ in $E_{\lambda}$, and so \[ u_{n}^{i}\to u_0^{i}\quad \text{in } H^2(\mathbb{R}^{N}). \] Using \eqref{3.1} and the constants $\kappa, \eta$ are independent of $\lambda_{n}$, we have \[ \frac{1}{2} \int_{\Omega}\left(|\Delta u_0^1|^2 +|\nabla u_0^1|^2\right)dx -\int_{\Omega}F(x, u_0^1)dx-\frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)|u_0^1|^{p}dx \geq \eta>0 \] and \[ \frac{1}{2} \int_{\Omega}\left(|\Delta u_0^2|^2+|\nabla u_0^2|^2\right)dx-\int_{\Omega}F(x, u_0^2)dx -\frac{\mu}{p}\int_{\mathbb{R}^N}\xi(x)|u_0^2|^{p}dx\leq \kappa <0, \] which implies that $u_0^{i}\neq 0$ and $u_0^1\neq u_0^2$. 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