\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 190, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/190\hfil Two types of ground state solutions]
{Two types of ground state solutions for a periodic Schr\"odinger
 equations with zero on the boundary of the spectrum}

\author[D. Qin, X. Tang \hfil EJDE-2015/190\hfilneg]
{Dongdong Qin, Xianhua Tang}

\address{Dongdong Qin \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{qindd132@163.com}

\address{Xianhua Tang (corresponding author) \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{tangxh@mail.csu.edu.cn}

\thanks{Submitted March 2, 2015. Published July 20, 2015.}
\subjclass[2010]{35J10, 35J20}
\keywords{Schr\"odinger equation; spectrum point zero; superlinear;
\hfill\break\indent  ground states; Nehari-Pankov manifold}

\begin{abstract}
 This article concerns the Schr\"odinger equation
 \begin{gather*}
 -\Delta u+V(x)u=f(x, u), \quad \text{for } x\in\mathbb{R}^N,\\
 u(x)\to 0, \quad \text{as } |x| \to \infty\,.
 \end{gather*}
 Assuming that $V$ and $f$ are periodic in $x$, and $0$ is a boundary point
 of the spectrum $\sigma(-\Delta+V)$, two types of ground state solutions
 are obtained with some super-quadratic conditions. 
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

 \section{Introduction and main results}

We consider the  Schr\"odinger equation
 \begin{equation} \label{ps}
\begin{gathered}
 -\Delta u+V(x)u=f(x, u), \quad \text{for } x\in\mathbb{R}^N,\\
 u(x)\to 0, \quad \text{as } |x| \to \infty,
 \end{gathered}
\end{equation}
 where $V:\mathbb{R}^N\to \mathbb{R}$ and $f:\mathbb{R}^N\times\mathbb{R}\to \mathbb{R}$ is superlinear as $|u|\to \infty$. \eqref{ps} has been widely investigated in the literature over the past several decades for both its importance in applications and mathematical interest, see, e.g., \cite{BD1, BJ, CR, D2, D3, D4, FTZD1, FTZD2, FTZD3, KS, L1, LS,LW, LWZ,M, P,Q,Q2,Q3, R, SW,SW2, T2, T1,T3,T4,T5,W, WZ, Y, YCD,ZZ} and the references therein. In this paper, we mainly study the case that $V$ and $f$ are periodic in $x$ and $0$ is a boundary point of the spectrum $\sigma(-\Delta+V)$, i.e.
\begin{itemize}
\item[(V1)]  $V\in C(\mathbb{R}^N, \mathbb{R})$ is 1-periodic in each of
$x_1, x_2, \ldots, x_N$, $0\in \sigma(-\Delta +V)$,
 and there exists a $b_0>0$
 such that $(0, b_0]\cap \sigma(-\Delta +V)=\emptyset$;
 \end{itemize}

 Compared to the situation that $0$ lies in a gap of $\sigma(-\Delta+V)$,
this case is very difficult because $H^1(\mathbb{R}^N)$ is no longer the
working space on which the variational functional
 associated with \eqref{ps} defines. Indeed, the working space is only a
Banach space, not a Hilbert space. In particular, there are considerably
fewer results, see \cite{BD1,M,Q2,Q3,T3, WZ, YCD}.


 Bartsch and Ding \cite{BD1} obtained a weak solution with a stronger
version of the classic condition
\begin{itemize}
 \item[(AR)]  there exist constants $\mu >2$ such that
 $$
 0<\mu F(x, t)\le tf(x, t), \quad \forall  (x, t)\in \mathbb{R}^N\times
 (\mathbb{R}\setminus \{0\}).
 $$
 \end{itemize}
Their main idea is to use an approximation argument to construct some kind
 of Palais-Smale sequence and show that after translations a subsequence
converges in certain sense to a weak solution
 $u$ of \eqref{ps}. Later, this result was improved by Willem and Zou \cite{WZ} by
 using an generalized weak link theorem.
In a recent publication, Yang et al.\cite{YCD} established existence of one weak
 solution of \eqref{ps} with the following Nehari type assumption:
\begin{itemize}
\item[(Ne)]  $t\mapsto \frac{f(x,t)}{|t|}$ is strictly increasing on
$(-\infty,0)\cup(0,\infty)$.
\end{itemize}
Their main technique is the same as that used in \cite{BD1, WZ}.
Later, this result was improved by authors's recent paper \cite{Q3} by using
a generalized linking theorem established in \cite{KS,LS}.
The following condition  seems  to be necessary to obtain the existence
of one weak solution of \eqref{ps} in \cite{BD1, WZ, YCD}.
\begin{itemize}
\item[(F0)] there exist constants $c_0>0$, $2< \varrho <2^*$ such that
 $$
 tf(x, t)\ge c_0|t|^{\varrho}, \quad \forall  (x, t)\in \mathbb{R}^N\times \mathbb{R}.
 $$
 \end{itemize}

 However, (F0) it is a severe restriction, since it strictly controls the growth
of $f(x, t)$ as $|t|\to \infty$.
There are  functions which are superlinear at both zero and infinity, but do
 not satisfy the condition (F0). For example
$f(x, t)=at|t|^{\alpha-2}\ln(1+|t|^{1/N})$ with $a>0$ and
 $\alpha\in (2, 2^*-1/N)$.
 Recently, with the aid of a proper variational framework,
Tang \cite{T3} weakened (AR), (F0) and (Ne) to the following assumptions:
 \begin{itemize}
 \item[ (F1)] $f\in C(\mathbb{R}^{N+1}, \mathbb{R})$ is 1-periodic in each
of $x_1, x_2, \ldots, x_N$, and there exist constants
 $c_1, c_2>0$ and $2<\varrho\le p<2^*$ such that
 $$
 c_1\min\{|t|^{\varrho}, |t|^2\}\le tf(x, t)\le c_2(|t|^{\varrho}+|t|^{p}), \quad
\forall  (x, t)\in \mathbb{R}^N\times \mathbb{R};
 $$

 \item[(F2)]  $\lim_{|t|\to \infty} F(x, t)/t^2=\infty$,  a.e. $x\in \mathbb{R}^N$.

 \item[(DL)]  $\mathcal{F}(x, t):=\frac{1}{2}tf(x, t)-F(x, t)>0$ for all
$x\in \mathbb{R}^N, t\in \mathbb{R}\setminus \{0\}$, and there exist $r_0>0$,
 $c_0>0$ and $\sigma>\max\{1, N/2\}$ such that
 $$
 |f(x, t)|^{\sigma}\le c_0|t|^{\sigma}\mathcal{F}(x, t), \quad \forall
 (x, t)\in \mathbb{R}^{N}\times \mathbb{R}, \; |t|\ge r_0;
 $$

 \item[(Ta)]  there exists a $\theta_0\in (0, 1)$ such that
 $$
 \frac{1-\theta^2}{2}tf(x, t) \ge \int_{\theta t}^{t}f(x, s)\mathrm{d}s
=F(x, t)-F(x, \theta t),
 \quad  \forall  \theta\in [0, \theta_0], \; (x, t)\in \mathbb{R}^N\times \mathbb{R}.
 $$
 \end{itemize}
 Under basic assumptions (V1), (F1) and (F2), assuming moreover (DL) or (Ta) holds,
{\it least energy solutions} were obtained by Tang \cite{T3},
i.e. a nontrivial solution $u_0\in E$ such that $\Phi(u_0)=\inf_{\mathcal{M}}\Phi$,
where
 \begin{equation}\label{M}
 \mathcal{M} = \{u\in E\setminus \{0\} : \Phi'(u)=0 \},
 \end{equation}
 $E$ is the working space on which the energy functional $\Phi$ associated with
 \eqref{ps} defines. In recent paper \cite{M}, Mederski studied \eqref{ps} under (Ne)
by using the generalized Nehari manifold method due to Szulkin and
Weth \cite{SW,SW2}, moreover (F0) was weakened to a similar version of (F1) there.
 For multiple results of \eqref{ps}, we refer to papers \cite{BD1,M,Q2}.

 Condition (DL) was introduced by Ding and Lee \cite{D2}, and it is commonly used
instead of condition (AR), see for instance \cite{D4,Q,T1,Ta,T3} and the references
therein. Clearly, (AR) is much stronger than (DL), (F2) and (Ta).
On the other hand, (Ta) was introduced by Tang \cite{T2} and is weaker than
the following mild version of (Ne):
\begin{itemize}
\item[(WN)]   $t\mapsto\frac{f(x,t)}{|t|}$ is non-decreasing on
$(-\infty,0)\cup(0,\infty)$.
\end{itemize}
 In addition, (F1) weakens (F0) greatly and is satisfied by many functions,
such as $f(x, t)=at|t|^{\alpha-2}\ln(1+|t|^{1/N})$ with $a>0$ and
$\alpha\in (2, 2^*-1/N)$. We point out that Liu \cite{L1} first uses (WN)
to replace of (Ne) for the case that $0$ lies in a gap of $\sigma(-\Delta+V)$,
and obtains a least energy solution. Later, this result is improved by
Tang in \cite{T2} by taking advantage of (Ta).

 It is well known that for periodic potential $V$, the operator
$\mathcal{A}:=-\Delta +V$ has purely continuous spectrum $\sigma (\mathcal{A})$
which is bounded below and consists of closed disjoint intervals
(see \cite[Theorem XIII.100]{RS}).
 Before stating our main results, we first present the following weaker
version of (DL) which was  introduced by Tang in recent paper \cite{T5}:
\begin{itemize}
\item[(F3)] $\mathcal{F}(x,t)\ge0$, and there exist constants $c_3>0$,
 $\delta_0\in(0,\Lambda_0)$ and $\sigma>\max\{1,N/2\}$ such that
 $$
 \frac{f(x,t)}{t}\ge(\Lambda_0-\delta_0)  \Longrightarrow
 \big(\frac{f(x,t)}{t}\big)^{\sigma}\le c_3\mathcal{F}(x,t),
 $$
 where $\Lambda_0:=\inf[\sigma(-\Delta +V)\cap (0, \infty)]$.

\end{itemize}
 Clearly, $\Lambda_0\ge b_0$ by (V1), and (DL) {\it implies} (F3) {\it under} (F1).
 More precisely, (F3) holds under (DL) and basic conditions that
$f\in C(\mathbb{R}^{N+1}, \mathbb{R})$ is 1-periodic in each of
$x_1, x_2, \ldots, x_N$ and $f(x,t)=o(|t|)$ as $|t|\to0$ uniformly in
$x\in\mathbb{R}^{N}$.
 Thus (F3) weakens (DL), and there are some functions satisfying (F3),
but not (DL), see Example \ref{examp1.7} and  \ref{examp1.8}. Moreover,
it is more convenient to use, see Lemma \ref{lem3.6}.


 In this article, we continue to study problem \eqref{ps}, and construct two
types of ground state solutions of \eqref{ps}, i.e. {\it the least energy solution}
and {\it the Nehari-Pankov type}. We first weaken (Ne) to (WN), and establish
the existence of a ground state solution of Nehari-Pankov type.
The {\it generalized Nehari manifold method} introduced by Szulkin and
Weth \cite{SW,SW2} can not be adopted due to the lack of strict monotonicity,
see Remark \ref{rmk1.4}.
So a new method is looked forward to being introduced which is the right
issue this paper intends to address. Inspired by the works
\cite{BD1, LWZ,P,R,SW,Ta,T3,T4,T5}, a more direct approach is used
in the present paper. The main ingredient in our approach is the observation
that a minimizing Cerami sequence for the energy functional can be found
outside the Nehari-Pankov manifold $\mathcal{N}^{-}$ by using the diagonal
method, see Lemma \ref{lem3.5}, part of which derives from recent papers of
Tang \cite{Ta,T4}. Moreover, under weaker condition (F3), a least energy
solution is obtained with the aid of a generalized linking theorem established
in \cite{T3}.


Let $E$, $E^-$ be the Banach space defined in Section 2. Under assumptions
(V1) and (F1), the  functional
 \begin{equation}\label{16}
 \Phi(u)=\int_{\mathbb{R}^N}(|\nabla u|^2+V(x)u^2) \mathrm{d}x
-\int_{\mathbb{R}^N}F(x,u) \mathrm{d}x,
 \end{equation}
 is well defined for all $u\in E$, moreover $\Phi\in C^1(E,\mathbb{R})$,
see Lemma \ref{lem2.2}. Denote the Nehari-Pankov manifold by
 \begin{equation}\label{Ne}
 \mathcal{N}^-=\{u\in E\setminus{E^-}: \langle\Phi'(u),u\rangle
=\langle\Phi'(u),v\rangle=0, \; \forall  v\in E^-\}.
 \end{equation}
The set $\mathcal{N}^-$ was  introduced by Pankov \cite{P}, which is a subset
of the Nehari manifold
 \begin{equation}\label{Ne-}
 \mathcal{N} = \{u\in E\setminus \{0\} : \langle \Phi'(u), u \rangle=0\}.
 \end{equation}

 Now, we are ready to state the main results of this article.


\begin{theorem} \label{thm1.1}
 Let {\rm (V1), (F1), (F2), (WN)}  be satisfied,  then  \eqref{ps} has a
solution $u_0\in E$ such that $\Phi(u_0)=\inf_{\mathcal{N}^-}\Phi\geq\kappa$, where
 $\kappa$ is a positive constant.
\end{theorem}

\begin{theorem} \label{thm1.2}
 Let {\rm (V1), (F1), (F2), (F3)}  be satisfied,  then  \eqref{ps} has a
solution $u_0\in E$ such that $\Phi(u_0)=\inf_{\mathcal{M}}\Phi$, where
$\mathcal{M}$ is defined by \eqref{M}.
\end{theorem}


Note that, $u\in\mathcal{N}^{-}$ if $u\neq0$ and $\Phi'(u)=0$. Hence
$\mathcal{N}^{-}$ contains all
 nontrivial critical points of $\Phi$, i.e. $\mathcal{M}$ is a very small
subset of $\mathcal{N}^-$.
In general, $\mathcal{N}^{-}$ contains infinitely many elements of $E$.
In fact, for any $u\in E^+\setminus \{0\}$, there exist $t=t(u)>0$
and $w=w(u)\in E^{-}$ such that $w+tu\in \mathcal{N}^{-}$ which is the
global maximum of $\Phi|_{E^{-}\oplus\mathbb{R}^{+}u}$, see Corollary \ref{coro2.6}
and Lemma \ref{lem3.3}. As a consequence of Theorem \ref{thm1.1}, the least energy value
$m:=\inf_{\mathcal{N}^-}\Phi$ has a minimax characterization given by
\[
 m=\Phi(u_0)=\inf_{u\in E^{+}\setminus \{0\}}
\max_{v\in E^{-}\oplus\mathbb{R}^{+}u}\Phi(v).
\]
 Note that this minimax principle is much simpler than the usual characterizations
related to the concept of linking. Since $u_0$ is a solution at which $\Phi$
 has least ``energy" in set $\mathcal{N}^{-}$, it was called a ground state
solution of Nehari-Pankov type in \cite{Ta,T4}.

We remark that Theorems \ref{thm1.1}  and \ref{thm1.2}   generalize and  improve the results
in \cite{BD1,M,Q2,T3, WZ, YCD}.

 As a motivation we recall the generalized Nehari manifold method introduced
by Szulkin and Weth \cite{SW,SW2}. For the case that $0$ lies in a gap
 of $\sigma(-\Delta+V)$, they obtained a ground state solution of Nehari-Pankov
type under (Ne) and some additional conditions. The generalized Nehari manifold
method developed there is based on a direct and simple reduction of the
strongly indefinite problem to a definite one. More precisely, a homeomorphism
between the Nehari-Pankov manifold $\mathcal{N}^-$ and a unit sphere $S^{+}$
in $E^{+}$ is established which allows to find a minimizing sequence on the
sphere and hence on the Nehari-Pankov manifold. We point out that the assumption
``strictly increasing" in (Ne) is very crucial in the argument of Szulkin
and Weth \cite{SW, SW2}. In fact, the starting point of their approach is to
show that for each $u\in E\setminus E^-$, the Nehari-Pankov manifold
$\mathcal{N}^-$ intersects $\hat{E}(u):=E^-\oplus \mathbb{R}^+u$ in exactly
one point $\hat{m}(u)$. The uniqueness of $\hat{m}(u)$ enables one to define
a map $u\mapsto \hat{m}(u)$, which is crucial to construct the homeomorphism
between $\mathcal{N}^-$ and $S^{+}$, see \cite[Chapter 4]{SW2}.

 \begin{remark} \label{rmk1.4}\rm
The ground state solution of Nehari-Pankov type can not be established by using
the generalized Nehari manifold method if (Ne)  was weakened to (WN).
 Indeed, without the strict monotonicity, the uniqueness of $\hat{m}(u)$
can not be guaranteed unless some additional conditions on the nonlinearity
are assumed, and so the homeomorphism between the Nehari-Pankov manifold
and the sphere can not be established
(see  \cite[(A.2) and Prop. 2.3]{SW} and \cite{ZZ}). Thus, it is infeasible
to find a minimizing sequence on the Nehari-Pankov manifold by reducing
the problem on the sphere which is a definite case. Compared to the
generalized Nehari manifold method, the approach used in this paper seems
more direct and simpler.
\end{remark}


 Before proceeding to the proof of Theorems \ref{thm1.1} and \ref{thm1.2}, 
we give some examples to  illustrate the assumptions.


 \begin{example} \label{examp1.5}\rm
$F(x, t)=h(x)|t|^2\ln (1+|t|^{1/N})$, where $h\in C(\mathbb{R}^N, (0,+\infty))$
is 1-periodic in each  of $x_1, x_2, \ldots$, $x_N$.
 It is not difficult to show that if
$\mathcal{{F}}(x,t)=1/(2N)|t|^{2+1/N}(1+|t|^{1/N})^{-1}\ge0$,
then $F$ satisfies (WN) and (F1)--(F3) with $\sigma>\max\{1,N/2\}$,
but it does not satisfies (AR) and (F0).
\end{example}

 \begin{example} \label{examp1.6}\rm
$F(x, t)=h(x)\min\{|t|^{\varrho_1}, |t|^{\varrho_2}\}$, where
$2<\varrho_1<\varrho_2<2^*$
 and $h\in C(\mathbb{R}^N, (0,+\infty))$ is 1-periodic in each
of $x_1, x_2, \ldots, x_N$.
 Clearly, (F1)--(F3) and (WN) hold for $F$ with
$\sigma=\rho_1/(\rho_1-2)>\max\{1,N/2\}$ , but (F0) fails.
\end{example}


\begin{example} \label{examp1.7}\rm
 $F(x,t)=2\sum^{m}_{i=1}|t|^{\beta_i}\sin^2(2\pi x_1)$, where $2^{*}>\beta_1
 >\beta_2>\cdots >\beta_m\ge 2$.
 It is easy to see that
$\mathcal{{F}}(x,t)=\sum^{m}_{i=1}(\beta_i-2)|t|^{\beta_i}\sin^2(2\pi x_1)\ge0$.
Then $F$ does not satisfies (AR) and (DL), but it satisfies (F3) with
$\sigma=\beta_1/(\beta_1-2)>\max\{1,N/2\}$.
\end{example}

 \begin{example} \label{examp1.8}\rm
$F(x,t)=a({8}/{5}|t|^{13/4}-4|t|^{11/4}+{9}/{2}|t|^{9/4})$, where $a>0$
and $N\le4$.
 By simple computation, one has $\mathcal{{F}}(x,t)=a|t|^{9/4}(\sqrt{
 |t|}-{3}/{4})^2\ge0$. Then $F$ does not satisfies (AR) and (DL),
but it satisfies (F3) with $\sigma=12/5$ if $a\in(0,8\Lambda_0/81)$.
\end{example}

 The remaining of this article is organized as follows.
In Section 2, some preliminary results are presented.
The proofs of main results will be given in the last Section.

 \section{Variational setting and preliminaries}

 In this section, as in \cite{T3}, we introduce the variational framework
associated with problem \eqref{ps}. Throughout this paper,
 we denote by $\|\cdot\|_s$ the usual $L^s(\mathbb{R}^N)$ norm for
$s\in [1,\infty)$ and  $C_i$, $i\in\mathbb{N}$ for different positive constants.
Let $\mathcal{A}=-\Delta+V$, then $\mathcal{A}$ is
 self-adjoint in $L^2(\mathbb{R}^N)$ with domain
$\mathfrak{D}(\mathcal{A})=H^2(\mathbb{R}^N)$.
Let $\{\mathcal{E}(\lambda):-\infty\le\lambda\le+\infty\}$
 be the spectral family of $\mathcal{A}$, and $|\mathcal{A}|^{1/2}$
be the square root of $|\mathcal{A}|$.
 Set $\mathcal{U}=id-\mathcal{E}(0)-\mathcal{E}(0-)$. Then $\mathcal{U}$
commutes with $\mathcal{A}$, $|\mathcal{A}|$
 and $|\mathcal{A}|^{1/2}$, and $\mathcal{A}=\mathcal{U}|\mathcal{A}|$
is the polar decomposition of
 $\mathcal{A}$ (see \cite[Theorem 4.3.3]{EE}).
Let $E_*=\mathfrak{D}(|\mathcal{A}|^{1/2})$,
 the domain of $|\mathcal{A}|^{1/2}$, then $\mathcal{E}(\lambda)E_*\subset E_*$
for all $\lambda\in \mathbb{R}$.
 On $E_*$ define an inner product
\begin{equation*}
 (u,v)_0=(|\mathcal{A}|^{1/2}u,|\mathcal{A}|^{1/2}v)_{L^2}+(u,v)_{L^2}, \quad
 \forall u,v\in E_*,
\end{equation*}
and the norm
\begin{equation*}
 \|u\|_0=\sqrt{(u,v)_0}, \quad  \forall  u\in E_*,
\end{equation*}
where and in the sequel, $(\cdot,\cdot)_{L^2}$ denotes the usual
$L^2(\mathbb{R}^N)$ inner product.


By  (V1), we can choose $a_0>0$ such that
 \begin{equation}\label{30}
 V(x)+a_0>0, \quad \forall  x\in \mathbb{R}^N.
 \end{equation}
For $u\in C^{\infty}_0(\mathbb{R}^N)$, one has
 \begin{equation}\label{31}
\begin{aligned}
 \|u\|_0^2
 & =  (|\mathcal{A}|u, u)_{L^2} +\|u\|_2^2 \\
&= ((\mathcal{A}+a_0)\mathcal{U} u, u)_{L^2}
 -a_0(\mathcal{U} u, u)_{L^2}+\|u\|_2^2 \\
 & \le  \|\mathcal{U} (\mathcal{A}+a_0)^{1/2}u\|_2\|(\mathcal{A}+a_0)^{1/2}u\|_2
 +a_0\|\mathcal{U} u\|_2\|u\|_2+\|u\|_2^2 \\
 & \le  \|(\mathcal{A}+a_0)^{1/2}u\|_2^2+(a_0+1)\|u\|_2^2 \\
 & \le  (1+2a_0+M)\|u\|_{H^1(\mathbb{R}^N)}^2
\end{aligned}
 \end{equation}
 and
 \begin{equation}\label{32}
\begin{aligned}
 \|u\|_{H^1(\mathbb{R}^N)}^2
 & \le  ((\mathcal{A}+a_0+1)u, u)_{L^2} \\
&= (\mathcal{A}u, u)_{L^2}+(a_0+1)\|u\|_2^2  \\
 & =  (\mathcal{U}|\mathcal{A}|^{1/2}u, |\mathcal{A}|^{1/2}u)_{L^2}+(a_0+1)\|u\|_2^2 \\
 & \le  \||\mathcal{A}|^{1/2}u\|_2^2+(a_0+1)\|u\|_2^2
 \le (1+a_0)\|u\|_{0}^2,
\end{aligned}
 \end{equation}
where $M=\sup_{x\in\mathbb{R}^N}|V(x)|$. Since $C^{\infty}_0(\mathbb{R}^N)$
is dense in $(E_*,\|\cdot\|_0)$ and $H^1(\mathbb{R}^N)$, thus
 \begin{equation}\label{33}
 \frac{1}{1+a_0}\|u\|^2_{H^1(\mathbb{R}^N)}\le \|u\|^2_0
\le (1+2a_0+M)\|u\|^2_{H^1(\mathbb{R}^N)}, \quad \forall
 u\in E_*=H^1(\mathbb{R}^N).
 \end{equation}

 Let
 \begin{equation*}
 E^{-}_*=\mathcal{E}(0)E_*,\quad
 E^{+}=[\mathcal{E}(+\infty)-\mathcal{E}(0)]E_*,
 \end{equation*}
 and
 \begin{equation}\label{34}
 (u,v)_*=\big(|\mathcal{A}|^{1/2}u,|\mathcal{A}|^{1/2}v\big)_{L^2},\quad
\|u\|_*=\sqrt{(u,u)_*}, \quad \forall  u,v\in E_*.
 \end{equation}



 \begin{lemma}[{\cite[Lemma 3.1]{T3}}] \label{lem2.1}
 Suppose that {\rm (V1)} is satisfied. Then $E_*=E^{-}_*\oplus E^{+}$,
 \begin{gather}\label{35}
 (u,v)_*=(u,v)_{L^2}=0, \quad \forall  u\in E^{-}_*, \; v\in E^{+}, \\
 \label{36}
 \|u^{+}\|^2_*\ge \Lambda_0\|u^{+}\|^2_{2}, \quad
 \|u^{-}\|^2_*\le a_0 \|u^{-}\|^2_2, \quad \forall
 u=u^{-}+u^{+}\in E_*=E^{-}_*\oplus E^{+},
 \end{gather}
where $b_0$ is given by {\rm (V1)}  and $a_0$ by \eqref{30}.
\end{lemma}

It is easy to see that $\|\cdot\|_*$ and $\|\cdot\|_{H^1(\mathbb{R}^N)}$
are equivalent norms on $E^{+}$, and if $u\in E_*$ then
 $u\in E^{+} \Leftrightarrow \mathcal{E}(0)u=0$.
Thus $E^{+}$ is a closed subset of $(E_*,\|\cdot\|_0)=H^1(\mathbb{R}^N)$.
We introduce a new norm on $E^{-}_*$ by setting
 \begin{equation}\label{313}
 \|u\|_{-}=(\|u\|^2_*+\|u\|^2_{\varrho})^{1/2}, \quad  \forall u\in E^{-}_*.
 \end{equation}
Let $E^{-}$ be the completion of $E^{-}_*$ with respect to $\|\cdot\|_{-}$.
Then $E^{-}$ is separable and reflexive,
$E^{-}\cap E^{+}=\{0\}$ and $(u,v)_*=0$, $\forall u\in E^{-}$, $v \in E^{+}$.
Let $E=E^{-}\oplus E^{+}$ and define norm $\|\cdot\|$ as follows
\begin{equation}\label{314}
 \|u\|=(\|u^{-}\|_{-}^2+\|u^{+}\|^2_*)^{1/2}, \quad
 \forall u=u^{-}+u^{+}\in E=E^{-}\oplus E^{+}.
\end{equation}
It is easy to verify that $(E,\|\cdot\|)$ is a Banach space, and
 \begin{equation}\label{315}
\sqrt{\Lambda_0}\|u^{+}\|_2\le \|u^{+}\|_*=\|u^{+}\|,\quad
\|u^{+}\|_s\le {\gamma}_s\|u^{+}\|,\quad \forall u\in E,\; s\in[2,2^*],
\end{equation}
where ${\gamma}_s\in (0,+\infty)$ is imbedding constant.

\begin{lemma}[{\cite[Lemma 3.2]{T3}}] \label{lem2.2}()
 Suppose that {\rm (V1)}  is satisfied. Then the following conclusions hold.
\begin{itemize}
\item[(i)]   $E^{-}\hookrightarrow L^s(\mathbb{R}^N)$ for $\varrho\le s\le 2^*$;

\item[(ii)]   $E^{-}\hookrightarrow H^1_{\rm loc}(\mathbb{R}^N)$ and
$E^{-}\hookrightarrow \hookrightarrow L^s_{\rm loc}(\mathbb{R}^N)$
for $2\le s<2^*$;

 \item[(iii)]  For $\varrho\le s\le 2^*$, there exists a constant $C_s>0$ such that
 \begin{equation}\label{316}
 \|u\|_{s}^s\le C_s\Big[\|u\|^s_*+\Big(\int_{\Omega}|u|^{\varrho}\mathrm{d}x
\Big)^{s/{\varrho}}
 +\Big(\int_{{\Omega}^c}|u|^2\mathrm{d}x\Big)^{s/2}\Big], \quad \forall u\in E^{-},
 \end{equation}
  where $\Omega\subset \mathbb{R}^N$ is any measurable set,
${\Omega}^c=\mathbb{R}^N\setminus{\Omega}$.
\end{itemize}
\end{lemma}

The following linking theorem is an extension of \cite{KS}
(see also \cite[Theorem 6.10]{W}),
 which plays an important role in proving our main results.


 \begin{proposition}[{\cite[Theorem 2.4]{T3}}]  \label{prop2.3}
 Let $X$ be real Banach space with $X=Y\oplus Z$, where $Y$
 and $Z$ are subspaces of $X$, $Y$ is separable and reflexive, and there exists
a constant $\zeta_{0}>0$ such that
 the following inequality holds
 \begin{equation}\label{201}
 \|P_1u\|+\|P_2u\|\le \zeta_{0}\|u\|, \quad \forall u\in X,
 \end{equation}
 where $P_1: X\to Y, \ P_2: X\to Z$  are the projections.
 Let $\{\mathfrak{f}_k\}_{k\in \mathbb{N}}\subset Y^*$  be the dense subset with
 $\|\mathfrak{f}_k\|_{Y^*}=1$,  and the  $\tau$-topology on $X$ be generated
by the norm
 \begin{equation}\label{202}
 \|u\|_{\tau}:=\max\Big\{\|P_2u\|, \sum^{\infty}_{k=1}\frac{1}{2^k}|
\langle\mathfrak{f}_k,P_1u\rangle|\Big\} , \quad \forall u\in X.
 \end{equation}
Suppose that the following assumptions are satisfied:
\begin{itemize}
\item[(H1)]   $\varphi \in C^1(X,\mathbb{R})$ is $\tau$-upper semi-continuous and
$\varphi':(\varphi_a,\|\cdot\|_{\tau}) \to (X^*,\mathcal{T}_{w^*})$ is continuous
for every $a\in \mathbb{R}$;

\item[(H2)]  there exists $r>\rho>0$ and $e\in Z$ with $\|e\|=1$ such that
 \begin{equation*}
 \kappa:=\inf\varphi(S_{\rho})>0\ge \sup\varphi(\partial Q),
 \end{equation*}
where
 \begin{equation*}
 S_{\rho}=\{u\in Z:  \|u\|=\rho\}, \quad
 Q=\{v+se:  v\in Y, \; s\ge 0, \; \|v+se\|\le r\}.
 \end{equation*}
\end{itemize}
 Then there exist $c\in[\kappa,\ \sup_{ Q}\varphi]$ and a sequence
$\{u_n\}\subset X$ satisfying
 \begin{equation}\label{23}
 \varphi(u_n)\to c, \quad \|\varphi'(u_n)\|_{X^*}(1+\|u_n\|)\to 0.
 \end{equation}
Such a sequence is called a Cerami sequence on the level $c$,
or a $(C)_c$-sequence.
\end{proposition}

Let $X=E$, $Y=E^{-}$ and $Z=E^{+}$. Then \eqref{201} is obvious true by
\eqref{314}. Since $E^{-}$ is separable
 and reflective subspace of $E$, then $(E^{-})^*$ is also separable.
Thus we can choose a dense subset
 $\{\mathfrak{f}_k\}_{k\in\mathbb{N}}\subset (E^{-})^*$ with
$\|\mathfrak{f}_k\|_{(E^{-})^*}=1$. Hence, it
 follows from \eqref{202} that
 \begin{equation}\label{332}
 \|u\|_{\tau}:=\max\Big\{\|u^{+}\|, \sum^{\infty}_{k=1}\frac{1}{2^k}|
\langle\mathfrak{f}_k,u^{-}\rangle|\Big\}, \quad \forall u\in E.
 \end{equation}
It is clear that
 \begin{equation}\label{333}
 \|u^{+}\|\le \|u\|_{\tau}\le \|u\|, \quad \forall u\in E.
 \end{equation}
By Lemma \ref{lem2.2}, it is easy to see that the functional $\Phi$ defined
by \eqref{16} is of class $C^1$, moreover
 \begin{equation}\label{334}
 \langle\Phi'(u),v\rangle=\int_{\mathbb{R}^N}(\nabla u\nabla v+V(x)uv)
\mathrm{d}x-\int_{\mathbb{R}^N}f(x,u)v \mathrm{d}x, \ \ \forall u,v \in E.
 \end{equation}
This shows that critical points of $\Phi$ are the solutions of \eqref{ps}.
Furthermore
 \begin{gather}\label{335}
 \Phi(u)=\frac{1}{2}(\|u^{+}\|^2-\|u^{-}\|^2_*)
-\int_{\mathbb{R}^N}F(x,u)\mathrm{d}x,\quad
\forall u=u^{+}+u^{-}\in E^{-}\oplus E^{+}=E,\\
\label{336}
\langle\Phi'(u),v\rangle=(u^{+},v)_*-(u^{-},v)_*
-\int_{\mathbb{R}^N}f(x,u)v \mathrm{d}x,\quad \forall u,v \in E.
 \end{gather}

\begin{lemma}[{\cite[Lemma 3.3]{T3}}]  \label{lem2.4}
 Suppose that {\rm (V1), (F1)}  are satisfied. Then $\Phi\in C^1(E,\mathbb{R})$
is $\tau$-upper semi-continuous and
 $\Phi':(\Phi_a,\|\cdot\|_{\tau})\to (E^*,\mathcal{T}_{w^*})$
is continuous for every $a\in \mathbb{R}$.
\end{lemma}

\begin{lemma} \label{lem2.5}
 Suppose that {\rm (V1), (F1), (WN)} are satisfied. Then
$$
\Phi(u)\geq\Phi(tu+w)+\frac{1}{2}\|w\|^2_* +\frac{1-t^2}{2}\langle\Phi'(u),u\rangle
-t\langle\Phi'(u),w\rangle,
$$
for all $u\in E$, $w\in E^-$, $t\geq0$.
\end{lemma}

The proof of the above lemma is the same as one of
 \cite[Lemma 2.4]{T4},  we omit it here.
From Lemma \ref{lem2.5}, we have the following two corollaries.

 \begin{corollary} \label{coro2.6}
 Suppose that {\rm (V1), (F1), (WN)}  are satisfied, assume moreover
$u\in \mathcal{N}^-$. Then
 \begin{equation}\label{3402}
 \Phi(u)\geq\Phi(tu+w), \quad  \forall w\in E^-,\; t\geq0.
 \end{equation}
\end{corollary}

 \begin{corollary} \label{coro2.7}
 Suppose that {\rm (V1), (F1), (WN)}  are satisfied. Then
 $$
\Phi(u)\geq\Phi(tu^+)+\frac{t^2\|u^-\|^2_*}{2}
 +\frac{1-t^2}{2}\langle\Phi'(u),u\rangle+t^2\langle\Phi'(u),u^-\rangle, \quad
\forall u\in E, \; t\geq0.
$$
\end{corollary}

\section{Proof of main results}


 \begin{lemma} \label{lem3.1}
 Suppose that {\rm (V1), (F1)} are satisfied. Then
 \begin{itemize}
\item[(i)]  there exists $\rho>0$ such that
\[
 m:=\inf_{\mathcal{N}^-}\Phi\geq \kappa:
=\inf\{\Phi(u):u\in E^+, \|u\|=\rho\}>0;
\]

\item[(ii)]  $\|u^+\|\geq\max\{\|u^-\|_*,\sqrt{2m}\}$ for all
 $u\in \mathcal{N}^-$;
\end{itemize}
\end{lemma}

\begin{proof}
 (i) The first inequality is a direct consequence of Corollary \ref{coro2.6},
since for every  $u\in \mathcal{N}^-$ there is $t>0$ such that
$\|t u^+\|=\rho$.
For any $\varepsilon>0$, (F1) implies the existence of $C_{\varepsilon}>0$
such that
 \begin{equation}\label{3-3}
 |F(x,t)|\le \varepsilon|t|^{2}+C_{\varepsilon} |t|^{p}, \quad
\forall (x,t)\in \mathbb{R}^N \times \mathbb{R},
 \end{equation}
which, together with \eqref{335} and Lemma \ref{lem2.2}, yields
 \begin{equation}\label{33b}
\begin{aligned}
 \Phi(u)&=\frac{1}{2}\|u\|^2-\int_{\mathbb{R}^N}F(x,u)\mathrm{d}x \\
 &\ge  \frac{1}{2}\|u\|^2-C_1(\varepsilon\|u\|^{\varrho}
 +C_{\varepsilon}\|u\|^{p}), \quad \forall u\in E^+.
\end{aligned}
 \end{equation}
Choosing an appropriate $\varepsilon$ we see that the second inequality
holds for some $\rho>0$.

 (ii) For $u\in \mathcal{N}^-$, it follows from (i), (F1) and \eqref{335} that
 \begin{equation*}
 m\le \frac{1}{2}(\|u^+\|^2-\|u^-\|_{*}^2)-\int_{\mathbb{R}^N}F(x,u)\mathrm{d}x
 \le \frac{1}{2}(\|u^+\|^2-\|u^-\|_{*}^2),
 \end{equation*}
hence, $\|u^+\|\ge \max\{\|u^-\|_{*},\sqrt{2m}\}$.
\end{proof}

\begin{lemma}[{\cite[Lemma 4.2]{T3}}] \label{lem3.2}
 Suppose that {\rm (V1), (F1), (F2)}  are satisfied.
Let $e\in E^+$ with $\|e\|=1$. Then there is a $r_1>0$ such that
 $\sup\Phi(\partial Q)\leq 0$ for $r\ge r_1$, where
 \begin{equation}\label{41}
 Q=\{w+se: w\in E^-, s\geq0, \|w+se\|\leq r\}.
 \end{equation}
\end{lemma}


\begin{lemma} \label{lem3.3}
Suppose that {\rm (V1), (F1), (F2), (WN)}  are satisfied. Then for any
 $u\in E^{+}\setminus \{0\}$,
$\mathcal{N}^{-}\cap (E^{-}\oplus \mathbb{R}^{+}u)\ne \emptyset$, i.e.,
there exist $t(u)>0$ and $w(u)\in E^{-}$ such that
$t(u)u+w(u)\in \mathcal{N}^{-}$.
\end{lemma}

\begin{proof}
By Lemma \ref{lem3.2}, there exists $R>0$ such that $\Phi(v)\le 0$ for
$v\in (E^{-}\oplus \mathbb{R}^{+} u)\setminus B_{R}(0)$.
By Lemma \ref{lem3.1} (i), $\Phi(tu)>0$ for small $t>0$. Thus,
$0<\sup\Phi(E^{-}\oplus \mathbb{R}^{+} u)<\infty$.
It is easy to see that $\Phi$ is weakly upper semi-continuous on
$E^{-}\oplus \mathbb{R}^{+} u$,
therefore, $\Phi(u_0)=\sup\Phi(E^{-}\oplus \mathbb{R}^{+} u)$ for some
$u_0\in E^{-}\oplus \mathbb{R}^{+} u$.
This $u_0$ is a critical point of $\Phi|_{E^{-}\oplus \mathbb{R} u}$,
so $\langle \Phi'(u_0), u_0 \rangle= \langle \Phi'(u_0), v \rangle=0$ for all
$v\in E^{-}\oplus \mathbb{R}\ u$. Consequently, $u_0\in \mathcal{N}^{-}\cap
 (E^{-}\oplus \mathbb{R}^{+} u)$.
\end{proof}

\begin{lemma} \label{lem3.4}
Suppose that {\rm (V1), (F1), (F2)} are satisfied.
 Then there exist a constant $c_0\in [\kappa, \sup \Phi(Q)]$ and a
sequence $\{u_n\}\subset E$ satisfying
 \begin{equation}\label{Ce1}
 \Phi(u_n)\to c_0, \quad  \|\Phi'(u_n)\|(1+\|u_n\|)\to 0,
 \end{equation}
 where $Q$ is defined by \eqref{41}.
\end{lemma}

The above lemma  is a direct corollary of Lemmas \ref{lem2.4}, \ref{lem3.1} (i), 
\ref{lem3.2} and  Proposition \ref{prop2.3}.


 \begin{lemma} \label{lem3.5}
Suppose that {\rm (V1), (F1), (F2), (WN)} are satisfied. Then there exist a
constant $c_*\in [\kappa,\ m]$ and a sequence  $\{u_n\}\subset E$ satisfying
 \begin{equation}\label{44}
 \Phi(u_n)\to c_*, \quad \|\Phi'(u_n)\|(1+\|u_n\|)\to 0.
 \end{equation}
\end{lemma}

 Using the diagonal method and taking into account Lemmas 
\ref{lem3.1}, \ref{lem3.2}, \ref{lem3.4} and
 Corollary \ref{coro2.6}, one can prove the above lemma by the same argument as
in the proof of \cite[Lemma 3.8]{Ta}.

 Lemma \ref{lem3.5} shows that a minimizing Cerami sequence for the energy functional
can be found outside the Nehari-Pankov manifold, from which one can easily
demonstrate a ground state solution of Nehari-Pankov type for problem \eqref{ps}.


\begin{lemma} \label{lem3.6}
Suppose that {\rm (V1), (F1), (F2), (F3)}  are satisfied. Then any sequence
 $\{u_n\}\subset E$ satisfying
 \begin{equation}\label{411}
 \Phi(u_n)\to c\ge 0,\quad \langle\Phi'(u_n),u^{\pm}_n\rangle\to 0,
 \end{equation}
is bounded in $E$.
\end{lemma}


\begin{proof}
The following argument is essentially contained in \cite[Lemma 3.5]{T5},
for the reader convenience we choose to write it in detail.
It follows from (F3) and \eqref{411} that
 \begin{equation}\label{L63}
 C_2\ge\Phi(u_n)-\frac{1}{2}\langle\Phi'(u_n),u_n\rangle
=\int_{\mathbb{R}^{N}}\mathcal{{F}}(x,u_n)\mathrm{d}x\ge 0.
 \end{equation}
First we prove that $\{\|u_n\|_*\}$ is bounded. To this end, arguing by
contradiction, suppose that $\|u_n\|_*\to \infty$.
Let $v_n=u_n/\|u_n\|_*$, then $\|v_n\|_*=1$. If
\begin{equation*}
 \delta:=\limsup_{n\to \infty}\sup_{y\in\mathbb{R}^N}
\int_{B(y,1)}|v^+_n|^2\mathrm{d}x=0,
\end{equation*}
then by Lions's concentration compactness principle \cite{L} or
 \cite[Lemma 1.21]{W}, $v^+_n\to 0$ in $L^s(\mathbb{R}^N)$ for $2<s<2^*$.
Set $\sigma'={\sigma}/({\sigma-1})$ and
 \begin{equation}\label{L64}
 \Omega_{n}:=\big\{x\in \mathbb{R}^N: \ \frac{f(x,u_n)}{u_n}
\le(\Lambda_0-\delta_0)\big\}.
 \end{equation}
Clearly, $2\sigma'\in(2,2^*)$ by the fact $\sigma>\max\{1,N/2\}$.
It follows from (F1) and \eqref{315} that
 \begin{equation}\label{L65}
\int_{{\Omega_n}}\frac{f(x,u_n)}{u_n}(v_n^{+})^2\mathrm{d}x
 \le (\Lambda_0-\delta_0)\|v^+_n\|_2^2
\le (1- \frac{\delta_0}{\Lambda_0})\|v^+_n\|_{*}^2
\le 1- \frac{\delta_0}{\Lambda_0}.
 \end{equation}
On the other hand, by virtue of (F3), \eqref{L63} and H\"{o}lder inequality,
one can get that
 \begin{equation}\label{L66}
\begin{aligned}
\int_{\mathbb{R}^N\setminus{\Omega_n}}\frac{f(x,u_n)}{u_n}(v_n^{+})^2\mathrm{d}x
&\le \Big[\int_{\mathbb{R}^N\setminus{\Omega_n}}
 (\frac{f(x,u_n)}{u_n})^{\sigma}\mathrm{d}x\Big]^{1/\sigma}
\|v^{+}_n\|^{2}_{2\sigma'} \\
&\le \Big[c_3\int_{\mathbb{R}^N\setminus{\Omega_n}}\mathcal{{F}}(x,u_n)
 \mathrm{d}x\Big]^{1/\sigma}  \|v^{+}_n\|^{2}_{2\sigma'} \\
&\le (c_3C_2)^{1/\sigma}\|v^{+}_n\|^{2}_{2\sigma'}=o(1).
\end{aligned}
\end{equation}
Combining \eqref{L65} and \eqref{L66}, and using (F1), \eqref{336} and
\eqref{411}, one has
 \begin{equation}\label{-416}
\begin{aligned}
 1+o(1)
 & =  \frac{\|u_n\|_{*}^2-\langle\Phi'(u_n),u_n^{+}-u_n^{-}\rangle}{\|u_n\|_{*}^2} \\
 & =  \int_{u_n\neq0}\frac{f(x,u_n)}{u_n}
[(v_n^{+})^2-(v_n^{-})^2]\mathrm{d}x \\
 & \le  \int_{\Omega_n}\frac{f(x,u_n)}{u_n}(v_n^{+})^2\mathrm{d}x
 +\int_{\mathbb{R}^N\setminus{\Omega_n}}\frac{f(x,u_n)}{u_n}(v_n^{+})^2\mathrm{d}x \\
 & \le  1- \frac{\delta_0}{\Lambda_0}+ o(1).
\end{aligned}
 \end{equation}
 This contradiction shows that $\delta>0$. The rest of the argument is the same
as in the proof of \cite[Lemma 4.4]{T3}.
\end{proof}

 Note that condition (WN) is stronger than (Ta), then one gets
directly the following lemma from \cite[Lemma 4.4]{T3}.


\begin{lemma} \label{lem3.7}
 Suppose that {\rm (V1), (F1), (F2),  (WN)}
are satisfied. Then any sequence $\{u_n\}\subset E$ satisfying \eqref{411}
 is bounded in $E$.
\end{lemma}

\begin{lemma}[{\cite[Corollary 2.3]{BD1}}] \label{lem3.8}
Suppose that {\rm (V1)} is satisfied. If $u\subset E$ is a weak solution
of the Schr\"odinger equation
 \begin{equation}\label{423}
 -\Delta u+V(x)u=f(x, u), \quad x\in\mathbb{R}^N,
 \end{equation}
 i.e.
 \begin{equation}\label{424}
 \int_{\mathbb{R}^N}(\nabla u\nabla \psi+V(x)u\psi)\mathrm{d}x
=\int_{\mathbb{R}^N}f(x,u)\psi\mathrm{d}x, \quad
\forall \psi\in C^{\infty}_0(\mathbb{R}^N),
 \end{equation}
then $u_n\to 0$ as $|x|\to \infty$.
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm1.1}]
 Lemma \ref{lem3.5} and \ref{lem3.7} imply the existence of a bounded sequence
$\{u_n\}\subset E$ satisfying \eqref{44}. By \eqref{315} and
Lemma \ref{lem2.2} (i), $\|u_n\|^{\varrho}_{\varrho}+\|u_n\|^{p}_{p}$ is also bounded. If
\begin{equation*}
 \delta:=\limsup_{n\to \infty}\sup_{y\in\mathbb{R}^N}\int_{B(y,1)}
|u_n^+|^2\mathrm{d}x=0,
\end{equation*}
then by Lions's concentration compactness principle, $u^+_n\to 0$ in
$L^s(\mathbb{R}^N)$ for $2<s<2^*$. From (F1), \eqref{335}, \eqref{336}
and \eqref{44}, one sees that
 \begin{align*}
 2c_{*}+o(1)
&= \|u_n^+\|^2-\|u_n^-\|^2_*-2\int_{\mathbb{R}^N}F(x,u_n)\mathrm{d}x \\
&\leq \|u_n^+\|^2=\int_{\mathbb{R}^N}f(x,u_n)u_n^+\mathrm{d}x+\langle\Phi'(u_n),u_n^+\rangle \\
&\leq c_2\int_{\mathbb{R}^N}(|u_n|^{\varrho-1}+|u_n|^{p-1})|u^+_n|\mathrm{d}x+o(1) \\
&\leq c_2(\|u_n\|_{\varrho}^{\varrho-1}\|u^+_n\|_{\varrho}+\|u_n\|_{p}^{p-1}
\|u^+_n\|_{p})\mathrm{d}x+o(1)=o(1)
\end{align*}
which is a contradiction. Thus $\delta>0$.

Going if necessary to a subsequence, we may assume the existence of
$k_n\in \mathbb{Z}^N$ such that
\[
 \int_{B(k_n,1+\sqrt{N})} |u^+_n|^2\mathrm{d}x>\frac{\delta}{2}.
\]
Let us define $v_n(x)=u_n(x+k_n)$ so that
 \begin{equation}\label{425}
 \int_{B(0,1+\sqrt{N})} |v^+_n|^2\mathrm{d}x>\frac{\delta}{2}.
 \end{equation}
Since $V(x)$ and $f(x,t)$ are periodic in $x$, we have $\|v_n\|=\|u_n\|$ and
 \begin{equation}\label{426}
 \Phi(v_n)\to c_*\in [\kappa,\ m], \quad \|\Phi'(v_n)\|(1+\|v_n\|)\to 0.
 \end{equation}
Passing to a subsequence, we have $v_n\rightharpoonup v_0$ in $E$,
$v_n\to v_0$ in $L^s_{\rm loc}(\mathbb{R}^N)$ for $2\leq s<2^*$ and
$v_n\to v_0$ a.e. on $\mathbb{R}^N$. \eqref{425} implies that $v_0\neq0$.
By a standard argument, one can prove that
 $\langle\Phi'(v_0),\psi\rangle=0$ for any $\psi\in C^{\infty}_0(\mathbb{R}^N)$.
It follows that $\Phi'(v_0)=0$ since $C^{\infty}_0(\mathbb{R}^N)$ is dense in $E$.
Then $v_0\in \mathcal{N}^-$ and so $\Phi(v_0)\geq m$.
On the other hand, by \eqref{426}, (WN) and Fatou's Lemma, we have
\begin{align*}
 m&\geq c_*=\lim_{n\to \infty}
\big[\Phi(v_n)-\frac{1}{2}\langle\Phi'(v_n),v_n\rangle\big]\\
&=\lim_{n\to \infty}\int_{\mathbb{R}^N}
 \big[\frac{1}{2}f(x,v_n)-F(x,v_n)\big]\mathrm{d}x \\
&\geq \int_{\mathbb{R}^N}\lim_{n\to \infty}
\big[\frac{1}{2}f(x,v_n)-F(x,v_n)\big]\mathrm{d}x\\
&=\int_{\mathbb{R}^N}\big[\frac{1}{2}f(x,v_0)-F(x,v_0)\big]\mathrm{d}x \\
&= \Phi(v_0)-\frac{1}{2}\langle\Phi'(v_0),v_0\rangle=\Phi(v_0).
\end{align*}
Then $\Phi(v_0)\leq m$ and so $\Phi(v_0)=m=\inf_{\mathcal{N}^-}\Phi\ge \kappa$
by Lemma \ref{lem3.1} (i). It follows from Lemma \ref{lem3.8}
 that $v_0$ is a ground state
solution of problem \eqref{ps}.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm1.2}]
Applying Lemmas \ref{lem3.4} and \ref{lem3.6}, there exists a bounded sequence
$\{u_n\}\subset E$ satisfying \eqref{Ce1}. Similar to the argument as in
the proof of Theorem \ref{thm1.1}, we can show that $\Phi'(\bar{u})=0$
for some $\bar{u}\in E\setminus\{0\}$, i.e. $\mathcal{M}\neq0$.
Let $\hat{c}:=\inf_{\mathcal{M}}\Phi$. By (F3), for any $u\in \mathcal{M}$,
one has
$$
\Phi(u)=\Phi(u)-\frac{1}{2}\langle\Phi'(u),u\rangle
=\int_{\mathbb{R}^N}\mathcal{{F}}(x,u)\mathrm{d}x\ge0;
$$
therefore $\hat{c}\ge0$.
Let $\{u_n\}\subset \mathcal{M}$ such that $\Phi(u_n)\to \hat{c}$.
Then $\langle\Phi'(u_n),v\rangle=0$ for any $v\in E$.
It follows from Lemma \ref{lem3.6} that $\{u_n\}$ is bounded in $E$.
The rest of the argument is the same as in the proof of Theorem \ref{thm1.1}
by using (F3) instead of (WN).
\end{proof}


\subsection*{Acknowledgments}
The authors would like to thank the anonymous referees their valuable comments
and suggestions. The first author wishes to thank the China Scholarship Council
for supporting his visit to the University of Nevada, Las Vegas.
This work is partially supported by the NNSF (No: 11171351)
and Hunan Provincial Innovation Foundation for Postgraduates (CX2015B037).



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