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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 25, pp. 1--26.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2018/25\hfil Periodic Schr\"odinger equations]
{Ground and bound states of periodic \\ Schr\"odinger equations with
super or asymptotically  linear terms}

\author[Q. Wu, D. Qin \hfil EJDE-2018/25\hfilneg]
{Qingfang Wu, Dongdong Qin}

\address{Qingfang Wu \newline
School of Traffic and Transportation Engineering,
Central South University,
Changsha, 410075 Hunan, China}
\email{212032@csu.edu.cn}

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

\thanks{Submitted October 4, 2017. Published January 18, 2018.}
\subjclass[2010]{35J20, 35J60, 35Q55}
\keywords{Schr\"odinger equations; minimax characterization;
\hfill\break\indent  perturbation method; Nehari-Pankov manifold; ground states}

\begin{abstract}
 This paper is concerned with existence of ground and bound states for
 a class of nonlinear Schr\"odinger equation with periodic potential.
 We impose general assumptions on the nonlinearity with super or 
 asymptotically  linear growth, and find some refinements of known 
 results and new results by using  the perturbation method and a mountain
 pass argument.  In particular, a critical point theory is established 
 for the asymptotically linear growth case.
\end{abstract}

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


 \section{Introduction and statement of main results}

We consider the nonlinear stationary Schr\"odinger equation
 \begin{equation} \label{ps}
\begin{gathered}
 -\triangle 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 $V\in C({\mathbb{R}}^{N})$ and $f\in C({{\mathbb{R}}^N}\times{\mathbb{R}})$ depend periodically
 on $x$.  Equation \eqref{ps} is a fundamental equation of quantum mechanics which
 appears in many applications, such as studies of Bose-Einstein condensates in
 condensed matter physics and gap solitons in photonic crystals \cite{P}.
 Solutions of \eqref{ps} can also be interpreted as stationary states of the
 corresponding reaction-diffusion equation
 which models phenomena from chemical dynamics \cite{BD1}.

The principal aim of this paper is to establish
the existence of ground and bound states of \eqref{ps} when $f$ is asymptotically
linear or superlinear as $|u|\to\infty$. Denote by $\Phi$ the energy functional
associated with \eqref{ps}, see \eqref{16}.
A  nontrivial solution $u_0$ of \eqref{ps} is called a ground state if $\Phi(u_0)$
is the lowest level for $\Phi$  at which there are nontrivial solutions of \eqref{ps},
i.e., $u_0$ satisfies
 \begin{equation}\label{ad17}
 \Phi(u_0)=\inf_{\mathcal{K}}\Phi, \quad \text{where }
 {\mathcal{K}}:=\{u\in E\setminus \{0\}: \Phi'(u)=0\},
 \end{equation}
it will be seen later that ground state $u_0$ can be characterized as the minimizer
of $\Phi$ on the  Nehari-Pankov manifold $\mathcal{N}^-$ which has infinite
dimension and infinite co-dimension, i.e.,
 \begin{equation}\label{17}
\begin{gathered}
 \Phi(u_0)=\inf_{\mathcal{N}^{-}}\Phi,\\
 \text{where }\mathcal{N}^-=\{u\in E\setminus E^-:
 \langle\Phi'(u),u\rangle=\langle\Phi'(u),v\rangle=0, \;\forall  v\in E^-\},
\end{gathered}
 \end{equation}
$\mathcal{N}^-$ is first introduced by Pankov \cite{P}, and $E, \ E^{-}$ are
working spaces defined later in Section 2. Since $V(x)$ is periodic in $x$,
the operator $\mathcal{A}:=-\triangle +V$
 has purely continuous spectrum $\sigma (\mathcal{A})$ which is bounded below
and consists of closed disjoint
 intervals (see \cite[Theorem XIII.100]{RS}). Depending on the location of $0$
in $\sigma (\mathcal{A})$,
 there are many results of existence and multiplicity solutions to equation \eqref{ps}
with periodic potential $V$,
 see, e.g., \cite{AL,BD1,CR,KS, M,P, Q1, R, SW,T2,T3,WZ,YCD, ZZ} for superlinear case
 and \cite{CT,D2,D3,D4,LS,LZ,J,Q4, SZ,T4, V2, V3} for asymptotically linear case.

 In this article, we  consider mainly the case when $0$ is a
 boundary point of the spectrum $\sigma(\mathcal{A})$, i.e. the potential
$V(x)$ satisfies the assumption
\begin{itemize}
\item[(H1)] $V\in C({\mathbb{R}}^{N})$ is 1-periodic in $x_{i}$, $i=1, 2, \dots, N$,
$0\in \sigma(\mathcal{A})$, and there exists $b_0>0$ such  that
$(0,b_0]\cap \sigma(\mathcal{A})=\emptyset$.
\end{itemize}
Different from the case $0\not\in\sigma(\mathcal{A})$, the working space is
 only a Banach space, not a Hilbert space.
 Another difficulty to overcome is the lack of a priori bounds
 for the Palais-Smale sequences or Cerami sequences. Existence of nontrivial
solutions were obtained in  \cite{BD1, YCD} with the aid of an approximation argument,
 and in \cite{WZ} using an improved generalized linking theorem.
 In recent paper \cite{T3}, a new variational setting which is more suitable
for this case was established by Tang.
 Moreover, ground state solutions satisfying \eqref{ad17} were obtained under
 following general assumptions:
\begin{itemize}
\item[(H2)] $f\in C({{\mathbb{R}}^N}\times{\mathbb{R}})$ 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
 \begin{equation}\label{13}
 c_1\min\left\{|t|^{\varrho},|t|^2\right\} \le tf(x,t)\le c_2(|t|^{\varrho} +|t|^{p}),
 \quad   \forall (x, t)\in \mathbb{R}^N\times \mathbb{R};
 \end{equation}

\item[(H3)]  $\lim_{|t|\to \infty}F(x, t)/t^2=\infty$ uniformly in
$ x\in \mathbb{R}^N$, where
 $F(x,t)=\int^{t}_0f(x,s)\mathrm{d}s$.
\end{itemize}
A similar result can be found in \cite{M} where following Nehari type condition
 was used,
\begin{itemize}
\item[(H4)]  $t\mapsto f(x,t)/|t|$ is strictly increasing on
$(-\infty,0)\cup(0,\infty)$.
\end{itemize}
The generalized Nehari manifold method used in \cite{M,SW} depends heavily on (H4),
 and it seems not valid
 by weakening (H4) to following weak version:
\begin{itemize}
\item[(H5)]  $t\mapsto f(x,t)/|t|$ is non-decreasing on
$(-\infty,0)\cup(0,\infty)$.
\end{itemize}
 Therefore it is interesting to obtain a ground state solution of \eqref{ps}
 without (H4). Using the non-Nehari manifold method introduced in \cite{T4},
existence of a ground state solution satisfying \eqref{17} was proved in \cite{Q1}
under (H2)--(H3) and (H5), however, it is still unknown whether the same result
can be found by using the generalized Nehari manifold method. So some new tricks
are looked forward to  being introduced which
 is the right issue this paper intends to address.
 When the nonlinearity $f$ is asymptotically linear at infinity,
 the variational framework and approaches used in \cite{BD1, M, WZ, YCD}
seem no more applicable,
 there seem only two papers in the literature dealing with this case, they are
references \cite{Q3, Q4}  where ground state solutions satisfying
 \eqref{ad17} and \eqref{17},
 were obtained using the concentration compactness arguments and the
non-Nehari manifold method.
 In particular, (H2), (H5) and following assumption on asymptotical behavior of
$f$ were used there.
\begin{itemize}
\item[(H6)]  $f(x, t)=V_{\infty}(x)t+f_{\infty}(x, t)$, where
 $V_{\infty}\in C(\mathbb{R}^N)$ is 1-periodic in $x_1$, $x_2$, \ldots, $x_N$,
 $f_{\infty}(x, t)=o(|t|)$
 as $|t|\to \infty$ uniformly in $x\in \mathbb{R}^{N}$, and
 $\inf V_{\infty}(x)>\overline{{\Lambda}}:=\inf[\sigma(\mathcal{A})\cap (0, \infty)]$.
\end{itemize}
 Therefore, it is nature to ask whether the asymptotically linear case can be
studied further via a new method different from those used in \cite{Q3, Q4}?

 Motivated by above works, in the paper, we shall introduce new tricks to solve
above intriguing problems. Using the idea introduced in \cite{Q5},
 we first establish a critical point theory for
 asymptotically linear growth case and show that $\inf_{\mathcal{N}^-}\Phi$ can
be attained by a nontrivial solution  of \eqref{ps}.
 Particularly, following weaker condition than (H4) is used, which allows to
introduce the Nehari-Pankov manifold and to define a ground state as minimizer
of the energy functional  on it.
\begin{itemize}
\item[(H7)]  $t\mapsto f(x,t)/|t|$ is non-decreasing on
$(-\infty,0)\cup(0,\infty)$,
  and strictly increasing on $[-\tau_0,0)\cup(0, \tau_0]$
  for some $\tau_0> 0$.
\end{itemize}
 Afterwards, we consider the superlinear growth case and introduce
 a abstract critical point theory on manifold $\mathcal{N}^{-}$ and following
manifold studied in \cite{BM2}:
 \begin{equation}\label{1n3}
 \mathcal{M}:=\{u\in E:  \Phi'(u)|_{E^{-}}=0 \}.
 \end{equation}

 Based on the perturbation method used in \cite{TQ1}, we first show that
$\inf_{\mathcal{N}^-}\Phi$ is achieved under (H2), (H3) and (H5).
Subsequently, we obtain a nontrivial solution of \eqref{ps} via a mountain pass
argument on  the constraint $\mathcal{M}$, provided the following classic
Ambrosetti-Rabinowitz condition  and convexity condition (H9) are satisfied.
 Minimax characterization of the corresponding critical value is also given if
 (H5) is additionally satisfied.
\begin{itemize}
\item[(H8)]  there exists $2<\mu<2^*$ such that
 \begin{equation}\label{qq16}
 tf(x,t)\ge\mu F(x,t), \quad \forall x\in \mathbb{R}^N, \; t\in\mathbb{R};
 \end{equation}

 \item[(H9)]  $F(x,t)$ is convex with respect to $t\in \mathbb{R}$ for all $x\in \mathbb{R}^N$,
or, equivalently, $F(x, t+s)-F(x, t)-f(x,t)s\ge 0$ for all $t, s\in \mathbb{R}$, $x\in \mathbb{R}^N$.
 \end{itemize}
 By Lemma \ref{lem3.4} we see that (H9) is satisfied if (H5) holds.

 Let $E$ be the Banach space defined in Section 2. Under assumptions (H1) and (H2),
 the following functional
 \begin{equation}\label{16}
 \Phi(u)=\frac{1}{2}\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}). A standard argument \cite{W} shows that
 critical points of $\Phi$ are the solutions of \eqref{ps}.


 Before stating result for asymptotically linear growth case,
 we introduce following condition:
 \begin{itemize}
 \item[(H10)]  $f(x, t)=V_{\infty}(x)t+f_{\infty}(x, t)$, where
 $V_{\infty}\in C(\mathbb{R}^N)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$
with $\inf V_{\infty}>0$,
 $f_{\infty}(x, t)=o(|t|)$ as $|t|\to \infty$ uniformly in $x\in \mathbb{R}^{N}$,
and there exists a $u_0\in E^{+}\setminus \{0\}$ such that
 \begin{equation}\label{P+}
 \|u_0\|_*^2-\|v\|_*^2-\int_{\mathbb{R}^N}V_{\infty}(x)(u_0+v)^2\mathrm{d}x<0, \quad
 \forall v\in E^{-}.
 \end{equation}
 \end{itemize}
The norm $\|\cdot\|_*$ is defined later in \eqref{34}.
 Condition (H10) is weaker than (H6). Indeed,
 let $\{\mathcal{E}(\lambda):-\infty<\lambda<+\infty\}$
 be the spectral family of operator $\mathcal{A}$,
 if $\inf V_{\infty}>\overline{{\Lambda}}$, then we have
$\bar{\Lambda}\|\bar{u}\|_2^2\le\|\bar{u}\|^2_{*}\le\bar{\mu}\|\bar{u}\|_2^2$ for
 every $\bar{u}\in(\mathcal{E}(\bar{\mu})-\mathcal{E}(0)) E\subset E^{+}$
by taking $\bar{\mu}\in \left(\bar{\Lambda},\inf V_{\infty}\right)$.
Thus for any $v\in E^{-}$,
\begin{align*}
 &  \|\bar{u}\|_*^2-\|v\|_*^2-\int_{\mathbb{R}^N}V_{\infty}(x)(\bar{u}+v)^2\mathrm{d}x\\
 &\le \|\bar{u}\|_*^2-\|v\|_*^2-\inf V_{\infty}\left(\|\bar{u}\|_2^2
 +\|v\|_2^2\right)\\
 &\le -\big[(\inf V_{\infty}-\bar{\mu})\|\bar{u}\|_2^2
+\inf V_{\infty}\|v\|_2^2\big]<0,
\end{align*}
 this shows \eqref{P+}.

 \begin{theorem} \label{thm1.1}
 Let {\rm (H1),  (H2), (H7), (H10)} be satisfied.
Then \eqref{ps} has a ground state solution $u_0\in E\setminus\{0\}$ satisfying
 $\Phi(u_0)=\inf_{\mathcal{N}^-}\Phi=\inf_{\mathcal{K}}\geq\kappa$, where
 $\kappa$ is a positive constant. Moreover,
 $$
 \int_{\mathbb{R}^N}\left[|\nabla u_0|^2+\left(V(x)-V_{\infty}(x)\right)u_0^2\right]
\,\mathrm{d}x<0.
 $$
\end{theorem}

 \begin{corollary} \label{coro1.2}
 Let {\rm (H1), (H2), (H6), (H7)}  be satisfied.
  Then there is a ground state solution $u_0\in E\setminus\{0\}$ of \eqref{ps}
such that  $\Phi(u_0)=\inf_{\mathcal{N}^-}\Phi=\inf_{\mathcal{K}}\Phi\geq\kappa>0$
and
 $$
 \int_{\mathbb{R}^N}\left[|\nabla u_0|^2+\left(V(x)-V_{\infty}(x)\right)u_0^2\right]
\,\mathrm{d}x<0.
 $$
\end{corollary}

Note that, $\mathcal{N}^{-}$ contains all
 nontrivial critical points of $\Phi$, i.e. the critical set $\mathcal{K}$ is a
very small subset  of $\mathcal{N}^-$.
Therefore $\inf_{\mathcal{N}^-}\Phi\le\inf_{\mathcal{K}}\Phi$. If there is a
 nontrivial solution $u_0$ such that $\Phi(u_0)=\inf_{\mathcal{N}^-}\Phi$,
then $\inf_{\mathcal{K}}\Phi$ is
 attained at $u_0$ and $\Phi(u_0)=\inf_{\mathcal{N}^-}\Phi=\inf_{\mathcal{K}}\Phi$
 which implies that $u_0$ is a ground state solution.

 For superlinear growth case, we have following results.

 \begin{theorem} \label{thm1.3}
 Let {\rm (H1), (H2)}  be satisfied.
\begin{itemize}
 \item[(a)] If {\rm (H3), (H5)}  hold, then \eqref{ps} has a solution
 $u_0\in E\setminus\{0\}$
 satisfying $\Phi(u_0)=\inf_{\mathcal{N}^-}\Phi=\inf_{\mathcal{K}}\Phi\ge\kappa_0$,
where $\kappa_0$ is a positive constant.

 \item[(b)] {\rm (H8), (H9)} hold, then \eqref{ps} has a nontrivial
 solution. Moreover if {\rm (H5)} is satisfied,
 then \eqref{ps} has a solution $u_0\in E\setminus\{0\}$ such that
 \begin{equation}\label{n3-1}
 \Phi(u_0)=\inf_{\mathcal{N}^-}\Phi
=\inf_{\gamma\in\Gamma}\sup_{t\in [0,1]}\Phi(\gamma(t))\ge \kappa_0>0,
 \end{equation}
  where
 $ \Gamma=\{\gamma\in C([0,1],\mathcal{M}): \gamma(0)=0,\; \|\gamma(1)^{+}\|> r, \;
 \Phi(\gamma(1))<0\}$,
 and $r>0$ satisfying $\inf_{ u\in E^{+}: \|u\|=r}\Phi(u)>0$.
 \end{itemize}
\end{theorem}

 Theorems \ref{thm1.1} and  \ref{thm1.3} give a positive answer to the problems 
mentioned in the introduction.
In Section 3, we give the proof of Theorem \ref{thm1.1} by improving the generalized
 Nehari manifold method \cite{BM,SW}, see Theorem \ref{thm3.1},
 as a consequence of which the least energy value
 $c_0:=\inf_{\mathcal{N}^-}\Phi$
 has a minimax characterization given by
 \begin{equation}\label{nq31}
 c_0=\Phi(u_0)=\inf_{v\in E_0^{+}\setminus \{0\}}
\max_{u\in E^{-}\oplus\mathbb{R}^{+}v}\Phi(u),
 \end{equation}
 where $E_0^+$ is defined later by \eqref{P+d}. Theorem \ref{thm1.1} complements the results
 in \cite{Q4} where Corollary \ref{coro1.2} was obtained via the concentration compactness
arguments.  Based on the generalized Nehari manifold method \cite{SW} and a
perturbation argument,  we find the Palais-Smale sequences in Section 4 and give
the proof of Theorem \ref{thm1.3} in Section 5, respectively. It is easy to see that
Theorem \ref{thm1.3}-(a) improves the related results in \cite{M, YCD} and
 Theorem \ref{thm1.3}-(b)
seems to be new. Compared with \cite[Theorem 1.1]{BD1},
the following necessary condition (H11) used in \cite{BD1, WZ, YCD} is not needed in 
Theorem \ref{thm1.3}-(b) with the additional
 convexity condition (H9).
 \begin{itemize}
\item[(H11)] there exist constants $C>0$, $2< \varrho <2^*$ such that
 $$
 tf(x, t)\ge C|t|^{\varrho}, \quad \forall (x, t)\in \mathbb{R}^N\times \mathbb{R}.
 $$
 \end{itemize}
 Under condition (H5), characterization of minimal energy can be derived from
Theorem \ref{thm1.3}-(b), i.e.,
 $$
 c_0=\Phi(u_0)
 =\inf_{v\in E^{+}\setminus \{0\}}\max_{u\in E^{-}\oplus\mathbb{R}^{+}v}\Phi(u)
 =\inf_{\gamma\in\Gamma}\sup_{t\in [0,1]}\Phi(\gamma(t)).
 $$
 Note that these minimax principles are much simpler than the usual characterizations
 related to the concept of linking.

 \begin{remark} \label{rmk1.4} \rm
 Under assumptions of Theorem \ref{thm1.1}, there exists a $\alpha_0>0$ such that
 \begin{equation}\label{a0}
\begin{gathered}
 f(x, t)f_{\infty}(x, t)\le0, \quad \forall (x, t)\in \mathbb{R}^N\times \mathbb{R}, \quad\text{and}\\
 f(x, t)f_{\infty}(x, t)<0\quad  \text{for } 0<|t|\le \alpha_0.
\end{gathered}
 \end{equation}
Indeed, by  (H2),  (H5) and (H10),  one sees that
 $f_{\infty}(x, t)/t$ is non-decreasing on $t\in (0,\infty)$ and non-increasing
 on $t\in (-\infty,0)$, and $f_{\infty}(x, t)/t\to -V_{\infty}(x)<0$ as
$|t|\to 0$. These  together with $f_{\infty}(x, t)=o(|t|)$ as $|t|\to \infty$
 imply that $tf_{\infty}(x, t)\le 0$ for all $(x, t)\in \mathbb{R}^N\times \mathbb{R}$.
Together with the fact $tf(x,t)>0$ for $t\neq0$ using \eqref{13},
we deduce that there exists a $\alpha_0>0$ satisfying \eqref{a0}.
\end{remark}

 Before proceeding to the proof of main results, we give some nonlinear examples.
Functions listed in Examples \ref{examp1.5}, \ref{examp1.6}
 satisfy all assumptions of  Corollary \ref{coro1.2}.
 For function $f$ defined by Example \ref{examp1.7}, it satisfies all the assumptions of
Theorem \ref{thm1.3}, but it does not satisfy (H11).


 \begin{example} \label{examp1.5}\rm
$f(x,t)=V_{\infty}(x)\min \{|t|^{\nu},1\}t$, where $\nu \in (0, 2^*-2)$,
$V_{\infty}\in C({\mathbb{R}}^{N})$
 is 1-periodic in each of $x_1, x_2, \ldots, x_N$ and
 $\inf V_{\infty}>\overline{\Lambda}$.
\end{example}

 \begin{example} \label{examp1.6}\rm
 $f(x, t)=V_{\infty}(x)t[1-\frac{1}{\ln (e+|t|^{\nu})}]t$,
 where $\nu\in (0, 2^*-2)$, $V_{\infty}\in C({\mathbb{R}}^{N})$ is 1-periodic in each
of $x_1, x_2, \ldots, x_N$ and $\inf V_{\infty}>\overline{\Lambda}$.
\end{example}

\begin{example} \label{examp1.7}\rm
$f(x, t)=h(x)\min\{\frac{1}{\varrho_1}|t|^{\varrho_1-2},
\frac{1}{\varrho_2} |t|^{\varrho_2-2}\}t$, where $2<\varrho_1<\varrho_2<2^*$
 and $h\in C(\mathbb{R}^N)$ is 1-periodic in each of $x_1, x_2, \ldots, x_N$ with
$\inf h>0$.
\end{example}

This article is organized as follows. In Section 2, we briefly introduce
the variational framework setting established  in \cite{T3}.
Equation \eqref{ps} with asymptotically linear growth nonlinearity is considered
in Section 3 where  a critical point theory is established and Theorem \ref{thm1.1}
 is proved.
 In Section 4, we consider the superlinear growth case and find the Palais-Smale
sequences on the manifolds  $\mathcal{M}$ and $\mathcal{N}^{-}$, respectively.
 Theorem \ref{thm1.3} is showed in the last Section by studying a perturbed problem
associated with \eqref{ps}.

\section{Variational setting}

 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.
Note that operator $\mathcal{A}=-\Delta+V$ is  self-adjoint in $L^2(\mathbb{R}^N)$
with domain $\mathfrak{D}(\mathcal{A})=H^2(\mathbb{R}^N)$.
Let $\{\mathcal{E}(\lambda):-\infty<\lambda<+\infty\}$  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 $\lambda\in \mathbb{R}$.
 $E_*$ is Hilbert space with the inner product
 \begin{equation*}
 (u,v)_0=\big(|\mathcal{A}|^{1/2}u,|\mathcal{A}|^{1/2}v\big)_{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*}
here and in the sequel, $(\cdot,\cdot)_{L^2}$ denotes the usual $L^2(\mathbb{R}^N)$
inner product.  Clearly, $C^{\infty}_0(\mathbb{R}^N)$ is dense in $E_*$.


By  (H1), there exists a $a_0>0$ such that
 \begin{equation}\label{30}
 V(x)+a_0>0, \quad \forall x\in \mathbb{R}^N.
 \end{equation}
The argument of \cite[(3.2)-(3.3)]{T3} shows that
 \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}
 Denote
\[
 E^{-}_*=\mathcal{E}(0)E_*,\quad
 E^{+}=[\mathcal{E}(+\infty)-\mathcal{E}(0)]E_*,
\]
 and
 \begin{equation}\label{34}
 (u,v)_*=\left(|\mathcal{A}|^{1/2}u,|\mathcal{A}|^{1/2}v\right)_{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 (H1)} is satisfied. Then $E_*=E^{-}_*\oplus E^{+}$,
 \begin{equation}\label{35}
 (u,v)_*=(u,v)_{L^2}=0, \quad \forall u\in E^{-}_*, \; v\in E^{+},
 \end{equation}
 and
 \begin{equation}\label{36}
 \|u^{+}\|^2_*\ge \overline{\Lambda} \|u^{+}\|^2_{2}, \quad
 \|u^{-}\|^2_*\le a_0 \|u^{-}\|^2_2, \quad
 \forall u=u^{-}+u^{+}\in E_*=E^{-}_*\oplus E^{+},
 \end{equation}
 where $a_0$ is given 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 $(E_*,\|\cdot\|_0)=H^1(\mathbb{R}^N)$.
 Define a new norm on $E^{-}_*$ by setting
 \begin{equation}\label{313}
 \|u\|_{-}=\left(\|u\|^2_*+\|u\|^2_{\varrho}\right)^{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,
 \begin{equation}\label{31+}
 E^{-}\cap E^{+}=\{0\}, \quad  (u,v)_*=0, \quad \forall u\in E^{-}, \; v \in E^{+}.
 \end{equation}
Set $E=E^{-}\oplus E^{+}$ and define norm $\|\cdot\|$ as follows
 \begin{equation}\label{314}
 \|u\|=\left(\|u^{-}\|_{-}^2+\|u^{+}\|^2_*\right)^{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{\overline{\Lambda}}\|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 (H1)} is satisfied. Then the following statements hold:
\begin{itemize}
\item[(i)]  $E^{-}\hookrightarrow L^s(\mathbb{R}^N)$ for $\varrho\le s\le 2^{*}$;

\item[(ii)]  $E^{-}\hookrightarrow H^1_{loc}(\mathbb{R}^N)$ and
$E^{-}\hookrightarrow \hookrightarrow L^s_{loc}(\mathbb{R}^N)$ 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}

 Since $E^{-}$ is a separable  and reflective subspace of $E$,
it follows that $(E^{-})^*$ is also separable. Thus one may choose a dense subset
 $\{\mathfrak{f}_k\}_{k\in\mathbb{N}}\subset (E^{-})^*$ with
$\|\mathfrak{f}_k\|_{(E^{-})^*}=1$. Now define a new norm
 \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}
The topology generated by $\|\cdot\|_{\tau}$ will be denoted by $\tau$ and
all topological notions related to  it will include the symbol. It is clear that
 \begin{equation}\label{333}
 \|u^{+}\|\le \|u\|_{\tau}\le \|u\|, \quad \forall u\in E.
 \end{equation}
If $\{u_n\}\subset E^{-}$ is bounded, then by \cite[Lemma 2.1]{T3},
 \begin{equation}\label{33201}
 u_n\xrightarrow{\tau} u\Longleftrightarrow u_n\rightharpoonup u.
 \end{equation}

 By (H2) and Lemma \ref{lem2.2}, 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, \quad \forall u,v \in E.
 \end{equation}
By \eqref{34}, \eqref{35} and \eqref{314}, it holds
 \begin{equation}\label{335}
\begin{aligned}
\Phi(u)
 &= \frac{1}{2}(\|u^{+}\|_{*}^2-\|u^{-}\|^2_*)-\int_{\mathbb{R}^N}F(x,u)\,\mathrm{d}x \\
 &= \frac{1}{2}(\|u^{+}\|^2-\|u^{-}\|^2_*)-\int_{\mathbb{R}^N}F(x,u)\,\mathrm{d}x
 ,\quad \forall u=u^{+}+u^{-}\in E,
\end{aligned}\end{equation}
 and
 \begin{equation}\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{equation}

\section{Critical point theory for asymptotically linear problems}

 Different from the superlinear linear case \cite{BM, M,SW}, the 
Nehari-Pankov manifold $\mathcal{N}^{-}$ is not homeomorphic to the
 unit sphere $S^{+}$ in $E^{+}$ for asymptotically linear problem \eqref{ps}.
 In this section, following the idea used in \cite{Q5}, a critical point theory for
 elliptic problem like or similar to \eqref{ps} is established in a Banach space.

To explain this in detail, we define a set
 \begin{equation}\label{P+d}
 E_0^{+}=\Big\{u\in E^{+}\setminus \{0\} : \|u\|^2-\|v^{-}\|^2
 -\int_{\Omega}V_{\infty}(x)| u+v|^2\mathrm{d}x<0, \;\forall v\in E^{-}\Big\}.
 \end{equation}
 Clearly, $E_0^{+}$ is nonempty by (H10) and it is a cone since for 
$v\in E_0^{+}$ and $\alpha\in \mathbb{R}\setminus\{0\}$ it holds $\alpha v\in E_0^{+}$.
 For any $u\in E\setminus E^{-}$, let
 $$
 E(u)=E^{-}\oplus \mathbb{R} u,\quad \hat{E}(u)=E^{-}\oplus \mathbb{R}^{+}u=E^{-}\oplus \mathbb{R}^{+} u^+.
 $$
 Define
 \begin{equation}\label{nq336}
 I(u)=\frac{1}{2}\|u^{-}\|^2_*+\int_{\mathbb{R}^N}F(x,u)\,\mathrm{d}x,
 \end{equation}
 then \eqref{335} can be rewritten as
 \begin{equation}\label{n335}
 \Phi(u)=\frac{1}{2}\|u^{+}\|^2-I(u),\quad \forall u=u^{+}+u^{-}\in E.
 \end{equation}
It will be shown later if (H2) and (H7) and (H10)  are satisfied, then 
$\Phi$ possesses the following properties:
\begin{itemize}
 \item[(H12)]  $I\in C^1(E,\mathbb{R})$ and $I(u)\ge I(0)=0$ for all $u\in E$;

 \item[(H13)]  $I$ is $\tau-$sequentially lower semicontinuous: 
 $u_n\xrightarrow{\tau} u\Rightarrow \liminf_{n\to\infty}I(u_n)\ge I(u)$;

 \item[(H14)]  if $u_n\xrightarrow {\tau} u$ and $I(u_n)\to I(u)$ then $u_n\to u$;

 \item[(H15)]  there exists a $r>0$ such that 
$ \kappa:=\inf_{ u\in E^{+},\|u\|=r}\Phi(u)>0$;

 \item[(H16)] For each $u\in (E_0^{+}\oplus E^{-})\setminus E^{-}$, there
 exists a unique nontrivial critical point $\hat{n}(u)$ of $J|_{\hat{E}(u)}$.
 Moreover $\hat{n}(u)$ is the unique global maximum of $J|_{\hat{E}(u)}$;
 \item[(H17)] For each compact subset 
$\mathcal{W}\subset (E_0^{+}\oplus E^{-})\setminus E^{-}$
 there exists a constant $C_{\mathcal{W}}>0$ such that
 $\|\hat{n}(u)\|\le C_{\mathcal{W}}$ for all $u\in \mathcal{W}$.
 \end{itemize}
 By (H16), we can define the mappings:
 \begin{equation}\label{qIu-}
 \hat{n}: (E_0^{+}\oplus E^{-})\setminus E^{-}\to\mathcal{N}^{-},\quad
 u\mapsto\hat{n}(u)\quad \text{and}\quad
 {n}:=\hat{n}|_{S_0^+}: S_0^+\to\mathcal{N}^{-},
 \end{equation}
 where
 $$
 S_0^+:=\{u\in E_0^{+} : \|u\|=1\}.
 $$
 For any $u\in (E_0^{+}\oplus E^{-})\setminus E^{-}$, we deduce 
from \eqref{n335}, (H12) and (H15)--(H17)
 that $\hat{E}(u)\cap\mathcal{N}^{-}=\left\{\hat{n}(u)\right\}$ and
 \begin{equation}\label{nl1}
 \frac{1}{2}\|\hat{n}(u)^{+}\|^2\ge \Phi(\hat{n}(u))
\ge \Phi(ru^{+}/\|u^{+}\|)\ge\kappa\Longrightarrow 
\|\hat{n}(u)^{+}\|\ge \sqrt{2 \kappa}.
 \end{equation}

\begin{theorem} \label{thm3.1}
 Suppose that {\rm (H12)--(H17)} are satisfied. Then following statements hold.
 \begin{itemize}
 \item[(a)]  $c_0:=\inf_{\mathcal{N}^{-}}\Phi\ge\kappa>0$ and 
$\Phi$ has a (PS)$_{c_0}$  sequence in $\mathcal{N}^{-}$.

 \item[(b)]  If $\Phi$ satisfies the (PS)$^{\tau}_{c_0}$ condition in 
$\mathcal{N}^{-}$, i.e.,
 every (PS)$_c$ sequence in $\mathcal{N}^{-}$ has a subsequence which 
converges in $\tau$, then $c_0$ is achieved by a critical point of $\Phi$.
 \end{itemize}
\end{theorem}

\begin{proof}  First, we claim that
\begin{itemize}
 \item[(i)] ${n}$ is a homeomorphism with inverse $\check{n}:
 \mathcal{N}^{-}\to S_0^+$, $z\mapsto z^+/{\|z^{+}\|}$;

 \item[(ii)] $ \Phi\circ n \in {C}^1(S^{+}_0,\mathbb{R})$ and
 \[
 \langle(\Phi\circ n) '(u),\phi\rangle
 = \|n(u)^+\|\langle \Phi'(n(u)),\phi\rangle,
\]
for all $\phi\in T_{u}(S^{+}_0) =\{\xi\in E^{+}\ | \ ( u,\xi)_{*}=0\}$,
where $T_{u}\left(S^{+}_0\right)$ is the tangent space of $S^{+}_0$ at $u$.

 \item[(iii)] if $\{u_n\}\subset S_0^+$ is a Palais-Smale sequence for
$ \Phi\circ n $, then $\left\{n(u_n)\right\}$
 is a Palais-Smale sequence for $\Phi$ in $\mathcal{N}^{-}$. 
If $\{z_n\}\subset \mathcal{N}^{-}$ is a bounded
 Palais-Smale sequence for $\Phi$, then $\left\{{n}^{-1}(z_n)\right\}$ 
is a Palais-Smale  sequence for $ \Phi\circ n $ in $S_0^+$;

 \item[(iv)] $u\in S_0^+$ is a critical point of $ \Phi\circ n $ if and only
 if $n(u)$ is a nontrivial  critical point of $\Phi$. 
Moreover, the corresponding critical values coincide and
 $\inf_{S^{+}_0} \Phi\circ n =\inf_{\mathcal{N}^{-}}\Phi$;

 \item[(v)] if $\Phi$ is even, then so is $ \Phi\circ n $.
 \end{itemize}

Statements (i)--(v) have been proved in \cite[Lemmas 3.4, 3.5 and Corollary 3.6]{Q5} 
in a Hilbert space.  Since the working space $E$ considered here is only a Banach 
space, the argument should be modified.  We postpone the proof to the Appendix.

 (a) By (i) and \eqref{nl1} we have 
$\Phi(z)\ge \Phi(rz^{+}/\|z^{+}\|)\ge\kappa$ for all $z\in \mathcal{N}^{-}
 \Longrightarrow \|z^{+}\|\ge \sqrt{2 \kappa}$. 
Thus $c_0\ge \kappa>0$. Existence of a
 (PS)$_{c_0}$ sequence for $\Phi$ in $\mathcal{N}^{-}$ follows from 
(ii), (iii) and the Ekeland variational
 principle \cite[Theorem 2.4]{W} since 
$c_0=\inf_{\mathcal{N}^{-}}\Phi=\inf_{S^{+}_0} \Phi\circ n $.

 To prove (b) we consider a (PS)$_{c}$ sequence $\{u_n\}\subset S^{+}_0$ 
for $ \Phi\circ n$.  Then $\{n(u_n)\}$ is a Palais-Smale sequence for $\Phi$ 
in $\mathcal{N}^{-}$ by (iii),  hence $n(u_n)\xrightarrow{\tau} z$ in 
$\mathcal{N}^{-}$ after passing to a subsequence.
 This implies that $u_n=n(u_n)^{+}/\|n(u_n)^{+}\|\to z^{+}/\|z^{+}\|$ and we
 have proved:
 \begin{itemize}
 \item[(vi)] if $\Phi$ satisfies the (PS)$^{\tau}_{c}$ condition in $\mathcal{N}^{-}$ 
for some $c>0$  then $ \Phi\circ n $ satisfies the (PS)$_{c}$ condition.
 \end{itemize}
Next observe that if $\Phi$ satisfies the (PS)$^{\tau}_{c_0}$ condition in 
$\mathcal{N}^{-}$,  then $c_0$ is achieved by a critical point $u\in S^{+}_0$ 
of $ \Phi\circ n $. It follows from (iv)
 that $n(u)\in \mathcal{N}^{-}$ is a critical point of $\Phi$ and $\Phi(n(u))=c_0$. 
This proves (b).
\end{proof}

 Note that, Theorem \ref{thm3.1} can also be used to consider other nonlinear problems 
for asymptotically linear case, such as  p-Laplacian problems, Dirac equation, 
Hamiltonian system and elliptic system.

\begin{lemma}[{\cite[Lemma 3.3]{T3}}] \label{lem3.2}
 Suppose that {\rm (H1), (H2)}  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 $a\in \mathbb{R}$, i.e.,
 \begin{gather*}
 u_n, u \in E, \; \|u_n-u\|_{\tau}\to 0\; \Rightarrow \;
\Phi(u) \ge \liminf_{n\to\infty}\Phi(u_n),\\
 u_n, u\in \Phi_a, \; \|u_n-u\|_{\tau}\to 0 \; \Rightarrow \;
 \lim_{n\to\infty}\langle\Phi'(u_n),v \rangle=\langle \Phi'(u),v\rangle, \;
 \forall\ v\in E,
 \end{gather*}
where $\Phi_a:=\{u\in E:\ \Phi(u)\ge a\}$ and $T_{w^*}$ denotes the weak$^{*}$ 
topology on $E^{*}$.
\end{lemma}


 \begin{lemma} \label{lem3.3} 
 Let {\rm (H1), (H2)}  be satisfied. Then {\rm (H12)--(H15)}  hold.
\end{lemma}
 
\begin{proof} It follows from (H2) and Lemma \ref{lem3.2} that (H12) and (H13) hold. 
Proof of (H15) is standard, see \cite[Lemma 3.1]{Q1}.
Next, we prove (H14).
 Assume that $u_n\xrightarrow{\tau} u$ and $I(u_n)\to I(u)$ for $u_n, \ u\in E$. 
Then $u^{+}_n\to u^{+}$ in $E^{+}$ and
 \begin{equation}\label{n6}
 \frac{1}{2}\|u_{n}^{-}\|^2_*+\int_{\mathbb{R}^N}F(x,u_{n})\,\mathrm{d}x \to 
\frac{1}{2}\|u^{-}\|^2_*+\int_{\mathbb{R}^N}F(x,u)\,\mathrm{d}x.
 \end{equation}
By the same argument as in \cite[Lemma 3.3]{T3}, we can show that 
$\{u^{-}_n\}\subset E^{-}$ is bounded. By \eqref{333}, 
$u^{-}_n\rightharpoonup u^{-}$  in $E^{-}$ and passing to a subsequence it holds
 $u_n\to u$ a.e. on $\mathbb{R}^N$.
 It follows from Fatou's Lemma and weakly lower semicontinuity of the norm that
 \begin{equation}\label{n7}
 \|u_{n}^{-}\|^2_*\to \|u^{-}\|^2_* \quad \text{and}\quad
 \int_{\mathbb{R}^N}F(x,u_{n})\,\mathrm{d}x\to \int_{\mathbb{R}^N}F(x,u)\,\mathrm{d}x.
 \end{equation}
Therefore $\|u_{n}^{-}-u^{-}\|^2_*= \|u_{n}^{-}\|^2_*+\|u^{-}\|^2_*
-2\langle u_{n}^{-},u^{-}\rangle_{*}\to 0$. Observe that
 \begin{equation}\label{n8}
\begin{aligned}
 \int_{\mathbb{R}^N} \left(F(x,u_n)-F(x,u_n-u)\right)\,\mathrm{d}x
 & =  \int_{\mathbb{R}^N}\int_0^1\frac{d}{dt}F(x,u_n-u+tu)\mathrm{d}t\,\mathrm{d}x \\
 & =  \int_0^1\int_{\mathbb{R}^N} f(x,u_n-u+tu)u\,\mathrm{d}x\mathrm{d}t.
\end{aligned}
 \end{equation}
By (H2) and H\"older inequality, for any $\Omega\subset \mathbb{R}^N$ it holds
 \begin{equation}\label{n9}
\begin{aligned}
 & \int_{\Omega} |f(x,u_n-u+tu)u|\,\mathrm{d}x \\
 & \le  c_2\int_{\mathbb{R}^N}\left(|u_n-u+tu|^{\varrho-1}
 +|u_n-u+tu|^{p-1}\right)|u\chi_{\Omega}|\,\mathrm{d}x \\
 & \le  c_2\|u_n-u+tu\|^{\varrho-1}_{\varrho}\|u\chi_{\Omega}\|_{\varrho}
 +c_2\|u_n-u+tu\|^{p-1}_{p}\|u\chi_{\Omega}\|_{p},
\end{aligned}
 \end{equation}
where $\chi_{\Omega}$ denotes the characteristic function on $\Omega$.
 By Lemma \ref{lem2.2}, $\{\|u_n-u+tu\|_{\varrho}\}_{n}$ and 
$\{\|u_n-u+tu\|_{p}\}_{n}$ are bounded. Then for any $\epsilon>0$ there 
is $\delta>0$ such that for any $\Omega$ with Lebesgue measure $|\Omega|<\delta$ 
it holds
 \begin{equation}\label{n10}
\int_{\Omega} |f(x,u_n-u+tu)u|\,\mathrm{d}x<\epsilon,\quad \forall n\in \mathbb{N}.
 \end{equation}
Moreover, for any $\epsilon>0$ there is $\Omega\in \mathbb{R}^N$ with $|\Omega|<\infty$ 
such that
 \begin{equation}\label{n11}
 \int_{\mathbb{R}^N\setminus \Omega} |f(x,u_n-u+tu)u|\,\mathrm{d}x<\epsilon,\quad
\forall n\in \mathbb{N}.
 \end{equation}
In light of \eqref{n10}, \eqref{n11} and Vitali convergence theorem, one sees that 
$f(x,tu)u\in L^{1}(\mathbb{R}^N)$ and
 \begin{equation}\label{n12}
 \int_{\mathbb{R}^N} f(x,u_n-u+tu)u\,\mathrm{d}x\to \int_{\mathbb{R}^N} f(x,tu)u\,\mathrm{d}x,\quad
\text{as } n\to\infty .
 \end{equation}
By \eqref{n8}, we have
 \begin{equation}\label{n13}
 \int_{\mathbb{R}^N} \left(F(x,u_n)-F(x,u_n-u)\right)\,\mathrm{d}x\to
 \int_0^1\int_{\mathbb{R}^N} f(x,tu)u\,\mathrm{d}x\mathrm{d}t
=\int_{\mathbb{R}^N} F(x,u)\,\mathrm{d}x,
\end{equation}
as $ n\to\infty$,
 this together with \eqref{n7} implies 
 \begin{equation}\label{n13b}
 \lim_{n\to\infty} \int_{\mathbb{R}^N} F(x,u_n-u)\,\mathrm{d}x=0.
 \end{equation}
By (H2), 
 \begin{equation}\label{n13+}
\begin{aligned}
 o(1)&= \int_{\mathbb{R}^N} F(x,u_n-u)\,\mathrm{d}x \\
 &\ge \frac{c_1}{\varrho}\Big(\int_{|u_n-u|\le 1}|u_n-u|^{\varrho}\,\mathrm{d}x
 +\int_{|u_n-u|\ge 1}|u_n-u|^{2}\,\mathrm{d}x\Big),
\end{aligned}
\end{equation}
then we deduce from \eqref{315}, \eqref{316} and the fact $u^{+}_n\to u^{+}$ 
in $E^{+}$ that
 \begin{align*}
&\|u_n^{-}-u^{-}\|_{\varrho}^\varrho\\
& \le  C_{1}\Big[\|u^{-}_n-u^{-}\|^\varrho_*
 +\int_{|u_n-u|\le 1}|u^{-}_n-u^{-}|^{\varrho}\,\mathrm{d}x
 +\Big(\int_{|u_n-u|\ge 1}|u^{-}_n-u^{-}|^2\,\mathrm{d}x\Big)^{\varrho/2}\Big] \\
& \le  C_{2}\Big[\|u^{-}_n-u^{-}\|^\varrho_*+\int_{|u_n-u|\le 1}|u^{+}_n
 -u^{+}|^{\varrho}\,\mathrm{d}x
 +\int_{|u_n-u|\le 1}|u_n-u|^{\varrho}\,\mathrm{d}x   \\
& \quad  +\Big(\int_{|u_n-u|\ge 1}|u^{+}_n-u^{+}|^2\,\mathrm{d}x
+\int_{|u_n-u|\ge 1}|u_n-u|^2\,\mathrm{d}x\Big)^{\varrho/2}\Big]  
  =  o(1).
 \end{align*}
This shows that $\|u_n-u\|\to 0$ as $n\to \infty$.

 Before proving (H16)--(H17), we introduce a useful result for functions 
satisfying (H5) or (H7).

 \begin{lemma} \label{lem3.4}
 Suppose that $h(x, t)$ is non-decreasing in $t\in \mathbb{R}$.
 \begin{itemize}
 \item[(i)]  If $h(x,0)=0$ for any $x\in \mathbb{R}^N$, then
 \begin{equation}\label{L31}
 \Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|
 \ge \int_{\theta\tau+\sigma}^{\tau}h(x, s)|s|\mathrm{d}s, \quad 
\forall \theta\ge 0, \; \tau,  \sigma\in \mathbb{R}.
 \end{equation}

 \item[(ii)]  If $h(x, t)=0\Leftrightarrow t=0$ for any $x\in \mathbb{R}^N$, then
 \begin{equation}\label{L32}
 \Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|
 > \int_{\theta\tau+\sigma}^{\tau}h(x, s)|s|\mathrm{d}s,
\end{equation}
 for all $\theta\ge 0$, $\tau\in \mathbb{R}$, $\sigma\in \mathbb{R}\setminus \{0\}$;
 moreover if $h(x, t)$ is strictly increasing on $[-\tau_0,0)\cup(0, \tau_0]$
for some $\tau_0> 0$, then
 \begin{equation}\label{L33}
 \frac{1-\theta^2}{2}|\tau|h(x, \tau)\tau
 > \int_{\theta\tau}^{\tau}h(x, s)|s|\mathrm{d}s, \quad 
 \forall \theta\in (0, 1), \; |\tau|\in (0, \tau_0].
 \end{equation}
 \end{itemize}
\end{lemma}

 
 \begin{proof} 
Since $h(x, t)$ is non-decreasing in $t\in \mathbb{R}$, then for any $x\in \mathbb{R}^N$,
 \begin{equation}\label{L34}
 h(x, s)\le h(x, \tau) \; \forall s \le \tau; \quad   h(x, s)\ge h(x, \tau), \;
 \forall s\ge \tau.
 \end{equation}
To show \eqref{L32}, we consider five possible cases. 
Since $sh(x, s)> 0$ for $s\neq0$, it follows from \eqref{L34} that
\smallskip

\noindent\textbf{Case 1.}  $\tau=0$,
 $$
 \int_{\theta\tau+\sigma}^{\tau}h(x, s)|s|\mathrm{d}s
= \int_{\sigma}^{0}h(x, s)|s|\mathrm{d}s
 < \Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|=0.
 $$
\smallskip

\noindent\textbf{Case 2.}  $0\le \theta\tau+\sigma\le \tau$ or
 $ \theta\tau+\sigma\le \tau<0$,
 \begin{align*}
 \int_{\theta\tau+\sigma}^{\tau}h(x, s)|s|\mathrm{d}s
 & \le  h(x, \tau)\int_{\theta\tau+\sigma}^{\tau}|s|\mathrm{d}s \\
 & =  \Big(\frac{1-\theta^2}{2}\tau^2-\theta\tau\sigma
 -\frac{\sigma^2}{2}\Big)h(x, \tau) \ \mathrm{sign}\tau \\
 & <  \Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|;
 \end{align*}
\smallskip

\noindent\textbf{Case 3.}  $\theta\tau+\sigma< 0 <\tau$,
 \begin{align*}
 \int_{\theta\tau+\sigma}^{\tau}h(x, s)|s|\mathrm{d}s
 &<  \int_0^{\tau}h(x, s)|s|\mathrm{d}s\le h(x, \tau)\int_0^{\tau}|s|\mathrm{d}s \\
 &\le  \frac{1}{2}\left(\tau^2-{\theta\tau}(\theta\tau+\sigma)- {\theta\tau\sigma}\right)h(x, \tau)  \\
 &=  \Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|;
 \end{align*}
\smallskip

\noindent\textbf{Case 4.} $\tau<0<\theta\tau+\sigma$,
\begin{align*}
 \int_{\tau}^{\theta\tau+\sigma}h(x, s)|s|\mathrm{d}s
 & >  \int_{\tau}^{0}h(x, s)|s|\mathrm{d}s
 \ge h(x, \tau)\int_{\tau}^{0}|s|\mathrm{d}s \\
 & \ge  \frac{1}{2}\Big(\tau^2-{\theta\tau}(\theta\tau+\sigma)
- {\theta\tau\sigma}\Big)h(x, \tau)  \\
 & =  -\Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|;
 \end{align*}
\smallskip

\noindent\textbf{Case 5.}  $0<\tau< \theta\tau+\sigma$ or 
$\tau< \theta\tau+\sigma\le0$,
 \begin{align*}
 \int_{\tau}^{\theta\tau+\sigma}h(x, s)|s|\mathrm{d}s
 & \ge  h(x, \tau)\int_{\tau}^{\theta\tau+\sigma}|s|\mathrm{d}s \\
 & =  \Big(\frac{\theta^2-1}{2}\tau^2+\theta\tau\sigma+\frac{\sigma^2}{2}\Big)
 h(x, \tau) \ \mathrm{sign}\tau \\
 & >  -\Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big)h(x, \tau)|\tau|.
 \end{align*}
 The above five cases show that \eqref{L32} holds. By the same argument, 
one can prove \eqref{L31}.
 
 Since $h(x, t)$ is strictly increasing on $[-\tau_0,0)\cup(0, \tau_0]$,
 then for $\tau\in (0, \tau_0]$,
 $$
 \int_{\theta\tau}^{\tau}h(x, s)|s|\mathrm{d}s < h(x, \tau) \int_{\theta\tau}^{\tau}|s|\mathrm{d}s
 =\frac{1-\theta^2}{2}|\tau|h(x, \tau)\tau, \quad \forall \theta\in (0, 1);
 $$
 Similarly for $\tau\in [-\tau_0, 0)$ we have
 \begin{align*}
 \int_{\tau}^{\theta\tau}h(x, s)|s|\mathrm{d}s
 &>  h(x, \tau) \int_{\tau}^{\theta\tau}|s|\mathrm{d}s\\
 &= \frac{1-\theta^2}{2}\tau^2h(x, \tau)
=-\frac{1-\theta^2}{2}|\tau|h(x, \tau)\tau, \quad 
\forall \theta\in (0, 1).
 \end{align*}
 Both cases show that \eqref{L33} holds.
 \end{proof}

 \begin{lemma} \label{lem3.5}  
 Let {\rm (H1), (H2)} be satisfied.
\begin{itemize}
 \item[(i)]  If {\rm (H5) }holds, then for any $u\in E$,
 \begin{equation}\label{0340}
 \Phi(u) \ge \Phi(tu+ v)+\frac{1}{2}\| v\|^2_*
 +\frac{1-t^2}{2}\langle\Phi'(u),u\rangle-t\langle\Phi'(u), v\rangle,
 \end{equation}
 for all $t\ge 0$, $v \in E^-$,
 and the inequality is strict for $v\in E^-\setminus\{0\}$, moreover if
$u\in \mathcal{N}^-$ then
 \begin{equation}\label{L3402}
 \Phi(u)>\Phi(tu+ v)+\frac{1}{2}\| v\|^2_*, \quad \forall t\geq0, \;
 v\in E^-\setminus\{0\}.
 \end{equation}

 \item[(ii)]  If {\rm (H7)} holds, then (i)  holds, and for any 
$u\in E\setminus\{0\}$,
 \begin{align*}% \label{n0340}
 \Phi(u)  > \Phi(tu) +\frac{1-t^2}{2}\langle\Phi'(u),u\rangle, \quad
\forall t\in (0, 1),\;  |u(x)|\le \tau_0,
 \end{align*}
moreover if $u\in \mathcal{N}^-$ then
 \begin{equation}\label{n3402}
 \Phi(u)> \Phi(tu), \quad \forall t\in (0, 1), \; |u(x)|\le \tau_0.
 \end{equation}
 \end{itemize}
\end{lemma}

 \begin{proof} 
 For any $x\in\mathbb{R}^N$,  from (H2), (H5) and \eqref{L32} it follows that
 \begin{equation}\label{nq6}
 \Big(\frac{1-\theta^2}{2}\tau-\theta\sigma\Big) f(x,\tau)
> \int_{\theta\tau+\sigma}^{\tau}f(x, s)\mathrm{d}s,
\end{equation}
 for all $\theta\ge 0$, $\tau\in \mathbb{R}$, $\sigma\in \mathbb{R}\setminus \{0\}$,
 Similarly, by (H7) and \eqref{L33} we have \eqref{nq6} and
 \begin{equation}\label{nq61}
 \frac{1-\theta^2}{2}\tau f(x,\tau)> \int_{\theta\tau}^{\tau}f(x, s)\mathrm{d}s,\quad
 \forall \theta\in (0, 1), \; |\tau|\in (0, \tau_0].
 \end{equation}
Let $u\in E\setminus\{0\}$. Since $ u\in L^{2}(\mathbb{R}^N)$, 
$\operatorname{meas} \{x\in \mathbb{R}^N: \ | u(x)|\le \tau_0\}=\infty$.
 Then we deduce from \eqref{336}, \eqref{nq336}--\eqref{n335} and 
\eqref{nq6}--\eqref{nq61} that
 \begin{equation}\label{n5}
\begin{aligned}
 &\Phi(u)-\Phi(tu+v)- \frac{1-t^2}{2}\langle \Phi'(u),u\rangle
 +t\langle \Phi'(u),v\rangle  \\
 &=  -\big[I(u)-I(tu+v)- \frac{1-t^2}{2}\langle I'(u),u\rangle
 +t\langle I'(u),v\rangle\big]  \\
 &= -\frac{1}{2}\|u^{-}\|^2_{*}+\frac{1}{2}\|tu^{-}+v\|^2_{*} 
 +\frac{1-t^2}{2}\|u^{-}\|^2_{*} -t\langle u^{-}, v\rangle_{*} \\
 &\quad +\int_{\mathbb{R}^N}\Big(\frac{1-t^2}{2}f(x,u)u-tf(x,u)v
 -\int^{u}_{tu+v}f(x,s)\,\mathrm{d}s\Big)\,\mathrm{d}x \\
 &=  \frac{1}{2}\|v\|^2_{*}+\int_{\mathbb{R}^N}\left(\frac{1-t^2}{2}f(x,u)u-tf(x,u)v
 -\int^{u}_{tu+v}f(x,s)\,\mathrm{d}s\right)\,\mathrm{d}x\\
\end{aligned}
 \end{equation}
which is positive when
 \begin{equation}\label{L601}
\begin{gathered}
   t\geq0 \text{ and } v\in E^-\setminus\{0\}, \text{ or when}\\
  t\in (0, 1), \; v=0 \text{ and } |u(x)|\le \tau_0.
 \end{gathered}
\end{equation}
 By the definition of $\mathcal{N}^-$ and \eqref{L601}, we have (i) and (ii).
\end{proof}

 \begin{lemma} \label{lem3.6} 
 Let {\rm (H1), (H10), (H2), (H7)} be satisfied. 
Then (H16)--(H17) hold.
\end{lemma}

 \begin{proof} 
 By the same argument as in \cite[Lemma 3.2]{Q4}, we can show that
 \begin{itemize}
 \item[(H18)] for any compact set $\mathcal{W}\subset (E_0^{+}\oplus E^{-})
\setminus E^{-}$,
 there is a $R>0$ such that $\Phi(\cdot)\le 0$ on $\hat{E}(u)\setminus B_R(0)$ 
for every $u\in \mathcal{W}$;
 \end{itemize}

To prove (H16), we first prove that $ \mathcal{N}^{-}\cap \hat{E}(u)\neq \emptyset$ 
for each  $u\in (E_0^{+}\oplus E^{-})\setminus E^{-}$. 
Since $\hat{E}(u)=\hat{E}(u^{+})$ for each
 $u\in (E_0^{+}\oplus E^{-})\setminus E^{-}$, we may assume that 
$u\in E_0^{+}$ and $\|u\|=1$.
 By \eqref{nq336}, \eqref{n335} and (H18) above, there exists $R>0$ such that 
$\Phi(\cdot)\le 0$ on  $E^{-}\cup (\hat{E}(u) \setminus B_R(0))$. 
\cite[Lemma 3.1]{Q1} (i.e. (H15)) yields that $\Phi(ru^{+})\ge \kappa>0$ for 
some $r>0$,  thus $0<\sup_{\hat{E}(u)}\Phi<\infty$. 
By Lemma \ref{lem3.2}, $\Phi$ is weakly upper semi-continuous
 on $\hat{E}(u)$, therefore $\Phi(u_0)=\sup_{\hat{E}(u)}\Phi$ for
 some $u_0:=t_0 u+v_0\in \hat{E}(u)\setminus \{0\}$. 
This $u_0$ is a critical point of
 $\Phi|_{\hat{E}(u)}$, so
 $$
 \langle \Phi'(u_0), u_0\rangle=\langle \Phi'(u_0), v\rangle=0,
 \quad \forall v\in E^{-}.
 $$
 Consequently, $u_0\in \mathcal{N}^{-}\cap \hat{E}(u)$, and $t_0>0$ 
(otherwise $\Phi(u_0)\le0$, a contradiction).
 
 Suppose that $u_1$, $u_2$ are two critical points of $\Phi|_{\hat{E}(u)}$. 
Then $u_i\in \mathcal{N}^{-}\cap \hat{E}(u)$ for $i=1,2$ and we deduce from 
\eqref{0340} that
 \begin{equation}\label{h81}
 \Phi(u_1)=\max_{\hat{E}(u)}\Phi(\cdot)= \Phi(u_2).
 \end{equation}
 Moreover, it holds $u_1=t_1 u_2+v_1$, $u_2=t_2 u_1+v_2$ with 
$v_i\in E^{-}$ and $t_i>0$.
 By \eqref{L3402} we see that $v_1=v_2=0$, i.e. $u_1=t_1 u_2$.
 Let $\Omega_i:=\{x\in \mathbb{R}^N: \ | u_i(x)|\le \tau_0\}$, $i=1,2$. 
Since $ u_i\in L^{2}(\mathbb{R}^N)$, the measure $|\Omega_i|=\infty$.
 If $t_1<1$, then by \eqref{n3402} we have
 \begin{equation*}\label{h811}
 \Phi(u_2)>\Phi(t_1u_2)=\Phi(u_1).
 \end{equation*}
which contradicts with \eqref{h81}. Similarly $t_1>1$ leads to 
$\Phi(u_2)<\Phi(u_1)$. Then we get $t_1=1$, i.e. $u_1=u_2$.
Hence for each
 $u\in (E_0^{+}\oplus E^{-})\setminus E^{-}$, 
$ \mathcal{N}^{-}\cap \hat{E}(u)$ consists of precisely one point
 denoted by $\hat{n}(u)$ which is the unique global maximum of 
$\Phi|_{\hat{E}(u)}$.

Condition (H17) follows immediately form (H15)--(H16) and (H18).
\end{proof}

 \begin{lemma} \label{lem3.7} 
 Let {\rm (H1), (H2), (H7), (H10)} be satisfied. Then each
 Palais-Smale sequence of $\Phi|_{\mathcal{N}^{-}}$ is bounded, 
i.e. any sequence  $\{u_n\}\subset \mathcal{N}^{-}$ satisfying
 \begin{equation}\label{h82+}
 \|\Phi'(u_n)\|_{E^*}\to 0, \quad  \Phi(u_n)\le d \quad \text{for some }
 d\in [c_0,\infty)
 \end{equation}
 is bounded in $E$
\end{lemma}


 \begin{proof} 
 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
 $$
 \delta:=\limsup_{n\to\infty}\sup_{y\in \mathbb{R}^N}\int_{B(y,1)}|v_n^{+}|^2\mathrm{d}x=0,
 $$
 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 $2<s<2^*$. Fix $R>[2(1+d)]^{1/2}$. It follows from (H2) that
 \begin{equation}\label{Ln61}
 \limsup_{n\to\infty}\int_{\mathbb{R}^N}F(x, Rv_n^{+})\mathrm{d}x
 \le \lim_{n\to\infty} c_2\left(R^{\varrho}\|v_n^{+}\|_{\varrho}^{\varrho}
+R^{p}\|v_n^{+}\|_{p}^{p}\right)=0.
 \end{equation}
Let $t_n=R/\|u_n\|_{*}$. Using \eqref{L3402} and \eqref{h82+}--\eqref{Ln61}, 
one has
 \begin{align*}
 d
 & \ge  \Phi(u_n)\\
 & \ge  \Phi(t_nu_n^{+})+\frac{t_n^2}{2}\|u_n^{-}\|_*^2
 = \Phi(Rv_n^{+})+\frac{R^2}{2}\|v_n^{-}\|_*^2\\
 & =  \frac{R^2}{2}\|v_n\|_*^2-\int_{\mathbb{R}^N}F(x, Rv_n^{+})\mathrm{d}x\\
 & =  \frac{R^2}{2}+o(1) > d+1+o(1),
 \end{align*}
 which is a contradiction. Thus $\delta>0$. Passing to a $\mathbb{Z}^N$-transformation 
and using the same argument as in \cite[Lemma 3.5]{Q4}, one can show that
 $\{\|u_n\|_{*}\}$ is bounded. 
By (H2) and $u_n\in \mathcal{N}^{-}$, we have
 \begin{equation}\label{Ln67}
\begin{aligned}
 \|u_n^{+}\|_*^2-\|u_n^{-}\|_*^2 
&= \int_{\mathbb{R}^N}f(x, u_n)u_n\mathrm{d}x\\
&\ge c_1\Big(\int_{|u_n|<1}|u_n|^{\varrho}\mathrm{d}x
 +\int_{|u_n|\ge 1}|u_n|^2\mathrm{d}x\Big).
\end{aligned}
 \end{equation}
Then we deduce from \eqref{315}--\eqref{316} that
 \begin{equation}\label{La68}
\begin{aligned}
 \|u_n^{-}\|_{\varrho}^{\varrho}
 & \le  C_3\Big[\|u_n^{-}\|_*^{\varrho}+\int_{|u_n|<1}|u_n^{-}|^{\varrho}\mathrm{d}x
 +\Big(\int_{|u_n|\ge 1}|u_n^{-}|^2\mathrm{d}x\Big)^{\varrho/2}\Big] \\
 & \le  C_4\Big[\|u_n^{-}\|_*^{\varrho}+\int_{|u_n|<1}|u_n^{+}|^{\varrho}\mathrm{d}x
 +\int_{|u_n|<1}|u_n|^{\varrho}\mathrm{d}x  \\
 & \quad +\Big(\int_{|u_n|\ge 1}|u_n^{+}|^2\mathrm{d}x
 +\int_{|u_n|\ge 1}|u_n|^2\mathrm{d}x\Big)^{\varrho/2}\Big] \le C_5.
\end{aligned}
 \end{equation}
 This shows that $\{\|u_n^{-}\|_{\varrho}\}_{n}$ is also bounded. 
By \eqref{314} we have the boundedness of $\{\|u_n\|\}$.
\end{proof}

 \begin{lemma}[{\cite[Corollary 2.3]{BD1}}] \label{lem3.8} 
Suppose that {\rm (H1)}  is satisfied. If $u\subset E$ is a weak solution of 
the Schr\"odinger equation
 \begin{equation}\label{423}
 -\triangle 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 $u(x)\to 0$ as $|x|\to \infty$.
\end{lemma}



\begin{proof}[Proof of Theorem \ref{thm1.1}]
 By Lemmas \ref{lem3.3}, \ref{lem3.6} and Theorem \ref{thm3.1}-(a), there is a sequence 
$\{u_n\}$ satisfying
 \begin{equation}\label{LT01}
 u_n\in \mathcal{N}^{-}, \quad  \Phi(u_n)\to c_0=\inf_{\mathcal{N}^{-}}\Phi,
 \quad  \| \Phi'(u_n)\|_{E^*}\to 0.
 \end{equation}
Applying Lemma \ref{lem3.7}, we see that $\{u_n\}$ is bounded in $E$, thus
 $\|u_n\|^{\varrho}_{\varrho}+\|u_n\|^{p}_{p}$ 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 $2<s<2^*$. We deduce from  (H2), \eqref{315}, \eqref{335}, \eqref{336} 
and \eqref{LT01} that
 \begin{align*}
 2c_0+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})+o(1)=o(1),
 \end{align*}
 which is a contradiction since $c_0>0$. Thus $\delta>0$.

 Going if necessary to a subsequence, we may assume the existence of
 $k_n\in \mathbb{Z}^N$ such that
 \begin{equation*}
 \int_{B(k_n,1+\sqrt{N})} |u^+_n|^2\,\mathrm{d}x>\frac{\delta}{2}.
 \end{equation*}
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 $f(x,u)$ are periodic in $x$, we have $\|v_n\|=\|u_n\|$ and
 \begin{equation}\label{426}
 \Phi(v_n)\to c_0, \quad \|\Phi'(v_n)\|_{E^*}\to 0.
 \end{equation}
Passing to a subsequence, we have $v_n\rightharpoonup v_0$ in $E$,
 $v_n\to v_0$ in $L^s_{loc}(\mathbb{R}^N)$ for $2\leq s<2^*$ and
 $v_n\to v_0$ a.e. on $\mathbb{R}^N$. \eqref{425} implies that $v^+_0\neq0$, 
so $v_0\neq0$. By a standard argument,
 we show that $\langle\Phi'(v_0),\psi\rangle=0$, for all 
$\psi\in C^{\infty}_0(\mathbb{R}^N)$. Since $C^{\infty}_0(\mathbb{R}^N)$
 is dense in $E$, we can conclude $\Phi'(v_0)=0$. 
Thus $v_0\in\mathcal{N}^-$ and $\Phi(v_0)\geq c_0$.
 On the other hand, by (H7), \eqref{335}, \eqref{336}, \eqref{426} and Fatou's 
Lemma, we have
 \begin{align*}
 c_0&= \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)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)v_n-F(x,v_n)\big]\,\mathrm{d}x
 =\int_{\mathbb{R}^N}\big[\frac{1}{2}f(x,v_0)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*}
 This shows $\Phi(v_0)\leq c_0$, thus $\Phi(v_0)=c_0=\inf_{\mathcal{N}^-}\Phi$, 
which together with Lemma \ref{lem3.8} implies that $v_0$ is a ground state solution 
of problem \eqref{ps}.
\end{proof}

\section{Palais-Smale sequence for the superlinear case}
 
 In this section, we consider the superlinear case and make use of 
following assumptions instead of (H16)--(H17).
  \begin{itemize}
 \item[(H19)]  $\|u^{+}\|+I(u)\to \infty$ as $\|u\|\to \infty$;
 \item[(H20)]  $I(t_nu_n)/t^2_n\to \infty$ if $t_n\to \infty$ and 
$u_n^{+}\to u^{+}$ for some $u^{+}\neq 0$ as $n\to \infty$;
 \item[(H21)]  For each $u\in \mathcal{N}^{-}$, $v\in E^{-}$ and $t\ge 0$,
 $$
 \frac{t^2-1}{2}\langle I'(u),u\rangle+t\langle I'(u),v\rangle+I(u)-I(tu+v)\le 0,
 $$
 the strict inequality holds if in addition $t\in (0,1]$ and $v\neq 0$, or 
$t\in (0,1)$, $v=0$  and $|u|\le \tau_0$;

 \item[(H21')] $\frac{t^2-1}{2}\langle I'(u),u\rangle+t\langle I'(u),v\rangle+I(u)
 -I(tu+v)< 0$ for every
 $u\in \mathcal{N}^{-}$, $t\ge 0$ and $v\in E^{-}$ such that $u\neq tu+v$;
 
\item[(H22)]  $I(u)<I(u+v)$ for every $ v\in E^{-}\setminus\{0\}$ and 
$u\in \mathcal{M}$, where 
 \begin{equation}\label{n3}
 \mathcal{M}=\{u\in E: \Phi'(u)|_{E^{-}}=0 \}=\{u\in E: I'(u)|_{E^{-}}=0 \}.
 \end{equation}
 \end{itemize}
 
 It follows from \eqref{L601} that (H21) is satisfied under (H7). 
By \cite[Proposition 4.2]{BM}, (H16)--(H17) hold for all $u \in E\setminus E^-$,
 under (H12), (H13), (H15), (H19), (H20) and (H21'). 
Combining the proof of Lemma \ref{lem3.6} with the argument of
 \cite[Proposition 4.2]{BM}, we see that (H16) and (H17) hold
for all $u \in E\setminus E^-$,
 also using (H21') instead of (H21).
 We deduce from the definition of $\mathcal{M}$ and \eqref{L601} that 
(H22) holds under the following strict convexity condition:
 \begin{itemize}
 \item[(H23)]  $F(x, t+s)-F(x, t)-f(x,t)s> 0$ for all $t, s\in \mathbb{R}, \ x\in \mathbb{R}^N$
 and $s\neq0$.
 \end{itemize}
 By \eqref{nq6} we see that (H23) holds if (H5) and (H2) are satisfied.

 In light of (H12) and (H22), $u\not \in E^{-}$ if $u\in \mathcal{M}\setminus\{0\}$. 
By the definition of $\mathcal{M}$ and
 a similar argument as in \cite[Proposition 4.2]{BM}, following conditions (H24)
and (H25) are satisfied under (H13), (H19), (H20) and (H22).
 \begin{itemize}
 \item[(H24)]  for any $u\in E^{+}$, there is a unique nontrivial critical point 
$m(u)$ of $\Phi|_{\{u\}\oplus E^{-}}$.
 Moreover, $m(u)$ is the unique global minimum of $\Phi|_{\{u\}\oplus E^{-}}$;

 \item[(H25)]  for each compact subset ${\mathcal{W}}\subset E^{+}$, 
there exists a constant $C_{\mathcal{W}}>0$
 such that $\|{m}(u)\|\le C_{\mathcal{W}}$ for all $u\in {\mathcal{W}}$.
 \end{itemize}

 
 Similarly, we can define mapping 
 \begin{equation}\label{qnu-}
 {m}: E^{+} \to\mathcal{M},\quad  u\mapsto {m}(u).
 \end{equation}
The following critical point theorem was established in \cite{BM2}.

 \begin{theorem}[{\cite[Theorem 4.1, 4.3]{BM2}}] \label{thm4.1}
 Let $\Phi\in C^1(E,\mathbb{R})$ satisfy {\rm (H12)--(H15), \\ (H19), (H20)}.

\noindent (I)  If {\rm (H21)}  is satisfied, then $c_0=\inf_{\mathcal{N}^{-}}\Phi>0$ 
and $\Phi$ has a (PS)$_{c_0}$  sequence in $\mathcal{N}^{-}$.
 
\noindent (II)  If {\rm (H22)} is satisfied, then following statements hold.
 \begin{itemize}
 \item[(i)]  The mapping ${m}: E^{+} \to\mathcal{M}$ is a homeomorphism with
 inverse $\mathcal{M}\ni u\mapsto u^{+}\in E^{+}$.

 \item[(ii)]  $\Phi$ has a (PS)$_{\hat{c}_0}$ sequence in $\mathcal{M}$ and
 \begin{equation}\label{n3b}
 \hat{c}_0=\inf_{\gamma\in\Gamma}\sup_{t\in [0,1]}\Phi(\gamma(t))>0,
 \end{equation}
 where
 \begin{equation}\label{n4}
 \Gamma=\{\gamma\in C([0,1],\mathcal{M}): \gamma(0)=0,\; \|\gamma(1)^{+}\|> r,\; 
 \Phi(\gamma(1))<0\},
 \end{equation}
 and $r>0$ satisfying $\inf_{ u\in E^{+}: \|u\|=r}\Phi(u)>0$.

 \item[(iii)] If in addition {\rm (H21)}  is satisfied, then 
$\hat{c}_0\le c_0$, and if $\hat{c}_0$ is achieved by a critical point then 
$c_0=\hat{c}_0$ .
 \end{itemize}
\end{theorem}

The functional $\Phi\circ m$ has the classical mountain pass geometry. 
In fact, (H15) and (H22) imply that
 \begin{equation}\label{n5+}
 \Phi\circ m(u)\ge \Phi(u)\ge \kappa>0 \quad \text{for } u\in E^{+},\; \|u\|=r,
 \end{equation}
 moreover, for $u\in E^{+}\setminus\{0\}$ write $m(tu)=tu+v_{t}$ with 
$v_{t}\in E^{-}$ and
 set $u_{t}=u+\frac{1}{t}v_{t}=\frac{1}{t}m(tu)$, then by (H20),
\[
 \frac{1}{t^2}I (m(t u))= \frac{1}{t^2} I(tu_{t})\to \infty \quad
\text{as } t\to\infty,
\]
 this implies that
 \begin{equation}\label{n5+1}
 \Phi\circ m(t u)=\frac{t^2}{2}\|u\|^2-I(m(t u))\to -\infty ~~~\text{as}~~ t\to\infty.
 \end{equation}
 Then the classical mountain pass geometry follows. Set
 \begin{equation}\label{n4+}
 \Sigma:=\{\sigma\in C([0,1],E^{+}):\ \sigma(0)=0,\ \|\sigma(1)\|> r, ~\text{and}~ \ \Phi\circ m(\sigma(1))<0\},
 \end{equation}
 the mountain pass value of $\Phi\circ m$ can be written by
 \begin{equation}\label{n3+}
 \hat{c}_0=\inf_{\sigma\in\Sigma}\sup_{t\in [0,1]}\Phi\circ m(\sigma(t))\ge \kappa>0.
 \end{equation}

 \begin{remark} \label{rmk4.2}\rm
 By \eqref{n5+1}, there is $t_0>0$ such that $\Phi( m (t_0u))<0$ for any
 $u\in E^{+}\setminus\{0\}$. Therefore the path 
$\gamma_{u}(t):=m(tt_0u^{+})$, $t\in [0,1]$, lies
in $\Gamma$. Assuming that (H5)  holds, it follows from \eqref{L3402}
that for any $u\in \mathcal{N}^{-}$,
 \begin{equation}\label{n17+}
 \sup_{t\in [0,1]}\Phi(\gamma_{u}(t))\le \Phi(u) \Longrightarrow \hat{c}_0\le c_0.
 \end{equation}
If $\hat{c}_0$ is achieved by a critical point $u_0\in E$,
then one deduces from \eqref{n335} and \eqref{n3+} that $u_0^{+}\neq 0$ and 
$u_0\in \mathcal{N}^{-}$.
This yields $\hat{c}_0=\Phi(u_0)\ge c_0$, and so $\hat{c}_0=\Phi(u_0)= c_0$.
\end{remark}

\begin{lemma} \label{lem4.3} 
 Let {\rm (H1)--(H3)}  be satisfied. Then {\rm (H12)--(H15),  (H19), (H20)}
 hold. Moreover, {\rm (H21)}  is satisfied under {\rm (H7)}, and {\rm  (H22)}
  holds under {\rm (H23)}.
\end{lemma}

 \begin{proof}
 It follows from Lemma \ref{lem3.3} that (H12)--(H15) hold.
 In view of \eqref{L601},  (H23) implies (H22), and (H7) yields (H21).
 
To prove (H19), assume that $\|u_n\|\to \infty$. By \eqref{nq336},
 \begin{equation}\label{n14}
\begin{aligned}
&\|u_n^{+}\|+I(u_n) \\
 & = \|u_n^{+}\|+\frac{1}{2}\|u_n^{-}\|^2_*+\int_{\mathbb{R}^N}F(x,u_n)\,\mathrm{d}x \\
 & \ge \|u_n^{+}\|+\frac{1}{2}\|u_n^{-}\|^2_*
 +\frac{c_1}{\varrho}\Big(\int_{|u_n|<1}|u_n|^\varrho\,\mathrm{d}x
+\int_{|u_n|\ge1}|u_n|^2\,\mathrm{d}x\Big).
\end{aligned}
 \end{equation}
If $\|u_n\|_{*}\to \infty$, the conclusion holds. 
Suppose $\|u_n\|_{*}$ is bounded,
 then $\|u^{-}_n\|_{\varrho}\to \infty$. 
By \eqref{315} and \eqref{316}, one has
 \begin{equation}\label{n15}
\begin{aligned}
 \|u_n^{-}\|_{\varrho}^\varrho
 & \le  C_{6}\Big[\|u^{-}_n\|^\varrho_*+\int_{|u_n|< 1}|u^{-}_n|^{\varrho}
\,\mathrm{d}x
 +\Big(\int_{|u_n|\ge 1}|u^{-}_n|^2\,\mathrm{d}x\Big)^{\varrho/2}\Big] \\
& \le  C_{7}\Big[\|u^{-}_n\|^\varrho_*+\int_{|u_n|< 1}|u^{+}_n|^{\varrho}
 \,\mathrm{d}x
+\int_{|u_n|< 1}|u_n|^{\varrho}\,\mathrm{d}x  \\
 & \quad +\Big(\int_{|u_n|\ge 1}|u^{+}_n|^2\,\mathrm{d}x
 +\int_{|u_n|\ge 1}|u_n|^2\,\mathrm{d}x\Big)^{\varrho/2}\Big] \\
% \label{La15+}
& \le  C_{8}\Big[1+\int_{|u_n|< 1}|u_n|^{\varrho}\,\mathrm{d}x
 +2^{(\varrho-2)/2}\Big(\int_{|u_n|\ge 1}|u_n|^2\,\mathrm{d}x\Big)^{\varrho/2}\Big],
\end{aligned}
 \end{equation}
this and \eqref{n14} imply that $\|u_n^{+}\|+I(u_n)\to \infty$.

To show (H20), suppose that there are sequences $t_n\to \infty$ and 
$u_n^{+}\to u^{+}\neq 0$ such  that $I(t_nu_n)/t^2_n$ is bounded. 
By \eqref{n14} and \eqref{n15}, one sees that
 $\{\|u^{-}_n\|_{*}+\|u_n^{-}\|_{\varrho}\}_{n}$ is bounded. 
Passing to a subsequence,  we may assume that $u^{-}_n\rightharpoonup u^{-}$ 
in $E^{-}$. Let $\Omega:=\{x\in \mathbb{R}^N: \ u^{+}+u^{-}\neq 0\}$.
 If $|\Omega|>0$, then we deduce from (H3) and Fatou's lemma that
 \begin{align*}
 \frac{I(t_nu_n)}{t^2_n}=\frac{1}{2}\|u_n^{-}\|^2_*
+\int_{\mathbb{R}^N}\frac{F(x,t^2_nu_n)}{t^2_n}\,\mathrm{d}x\to\infty,
 \end{align*}
 a contradiction. Thus $|\Omega|=0$, i.e. $u^{+}=-u^{-}$ a.e. on $\mathbb{R}^N$. 
By \eqref{31+}, $(u^{+},u^{-})_{*}=0$. Then $u^{+}=0$ which is also a 
contradiction. Therefore $I(t_nu_n)/t^2_n\to \infty$.
\end{proof}

\section{Perturbation method and proof of Theorem \ref{thm1.3}}

 In this section, we give the proof of Theorem \ref{thm1.3} by considering a 
perturbed problem associated with \eqref{ps}.
 For any $\epsilon>0$, define $F_{\epsilon}(x, u)=F(x, u)+\epsilon|u|^{\varrho}$ and
 \begin{equation}\label{Phe1}
 \Phi_{\epsilon}(u) = \frac{1}{2}\|u^{+}\|^2-I_{\epsilon}(u), \quad  u\in E,
 \end{equation}
 where
 \begin{equation}\label{Ieu}
 I_{\epsilon}(u) 
= \frac{1}{2}\|u^{-}\|_{*}^2+\int_{\mathbb{R}^N}F_{\epsilon}(x,u)\mathrm{d}x.
 \end{equation}
 Similarly, we define
 \begin{equation}\label{Ne0}
 \mathcal{N}_{\epsilon}^{-} := \big\{u\in E\setminus E^{-} : 
\Phi_{\epsilon}'(u)|_{\hat{E}(u)}=0 \big\} ,\quad
 \mathcal{M}_{\epsilon}:=\{u\in E:  I_{\epsilon}'(u)|_{E^{-}}=0 \}.
 \end{equation}
 Let
 \begin{equation}\label{ce}
 c_{\epsilon}=\inf_{ u\in\mathcal{N}_{\epsilon}^{-}} \Phi_{\epsilon} (u),\quad
 \hat{c}_{\epsilon}=\inf_{\gamma\in\Gamma_{\epsilon}}\sup_{t\in [0,1]}
 \Phi_{\epsilon}(\gamma(t))=\inf_{\sigma\in\Sigma_{\epsilon}}
\sup_{t\in [0,1]}\Phi_{\epsilon}\circ m_{\epsilon}(\sigma(t)),
 \end{equation}
 where
 \begin{gather*} %\label{qn4}
 \Gamma_{\epsilon}=\{\gamma\in C([0,1],\mathcal{M}_{\epsilon}):
  \gamma(0)=0,\; \|\gamma(1)^{+}\|> r_0,\;
  \Phi_{\epsilon}(\gamma(1))<0\}, \\
 %\label{qn4+}
 \Sigma_{\epsilon}:=\{\sigma\in C([0,1],E^{+}): \sigma(0)=0,\;
 \|\sigma(1)\|> r_0,\;  \Phi_{\epsilon}\circ m_{\epsilon}(\sigma(1))<0\},
 \end{gather*}
 $r_0>0$ satisfies $\inf_{ u\in E^{+}, \ \|u\|=r_0}\Phi_{\epsilon}(u)>0$ 
for all $\epsilon\in (0, 1]$,  and $m_{\epsilon}: E^{+} \to\mathcal{M}_{\epsilon}$.
 
 Note that $F_{\epsilon}$ satisfies (H7) (actually (H4) is also satisfied) 
and (H23) provided that $F$ satisfies (H5) and (H9), respectively.
\end{proof}

 \begin{lemma} \label{lem5.1} 
 Let {\rm (H2), (H5)} be satisfied. Then for every $\epsilon\in (0, 1]$,
 there exists a $\kappa_0>0$ independent of $\epsilon$ such that 
$c_{\epsilon}\ge \kappa_0$.
\end{lemma}

 \begin{proof} 
 For any $u\in \mathcal{N}_{\epsilon}^{-}$, by a similar argument as in 
\eqref{L3402} it holds
 \begin{equation}\label{3402}
 \Phi_{\epsilon}(u)\geq\Phi_{\epsilon}(tu+v)+\frac{1}{2}\|v\|^2_*, \quad
 \forall v\in E^-, \; t\geq0.
 \end{equation}
In view of (H2) and \eqref{315},
 \begin{align*} %\label{Ff2}
 \Phi_{\epsilon}(u)
 & \ge \Phi_{\epsilon}(tu^{+})
 = \frac{t^2}{2}\|u^{+}\|-\int_{\Omega}F(x,tu^{+})\mathrm{d}x
 -\epsilon t^{\varrho}\|u^{+}\|^{\varrho}_{\varrho} \\
 & \ge  \frac{t^2}{2}\|u^{+}\|^{2}-\big(\frac{c_2}{\varrho}+1\big)
 t^{\varrho}\|u^{+}\|^{\varrho}_{\varrho}
 -\frac{c_2}{p}t^{p}\|u^{+}\|^{p}_{p} \\
 & \ge  \frac{t^2}{2}\|u^{+}\|^{2}-\big(\frac{c_2}{\varrho}+1\big)
 \gamma^{\varrho}_{\varrho}t^{\varrho}\|u^{+}\|^{\varrho}
 -\frac{c_2}{p}\gamma_{p}^{p}t^{p}\|u^{+}\|^{p}
 \quad  \forall u\in \mathcal{N}_{\epsilon}^{-},\; t\ge0 .
 \end{align*}
 Note that $2<\varrho\le p$. Thus for $\|tu^{+}\|$ small, there exists a 
$\kappa_0>0$ independent of $\epsilon$ such that $c_{\epsilon}\ge \kappa_0$.
\end{proof}

\begin{lemma} \label{lem5.2} 
Let {\rm (H2)}  and {\rm (H9)}  be satisfied. 
Then for every $\epsilon\in (0, 1]$, there exists a
 $\hat{\kappa}>0$ independent of $\epsilon$ such that 
$\hat{c}_{\epsilon}\ge \hat{\kappa}$.
\end{lemma}

 \begin{proof} 
 Since $F_{\epsilon}$ satisfies (H23), we see that (H22) holds for 
$I_{\epsilon}$.
 Then for any $u\in \mathcal{M}_{\epsilon}$,
 \begin{equation*}
 \Phi_{\epsilon}( u)\ge \Phi_{\epsilon}(u^{+}).
 \end{equation*}
 We deduce from (H2) and \eqref{315} that for any $u\in E^{+}$,
 \begin{equation}\label{Ff2}
\begin{aligned}
 \Phi_{\epsilon}( m_{\epsilon}(u))
 &\ge \Phi_{\epsilon}(u)=\Phi(u)-\epsilon\|u\|^{\varrho}_{\varrho} \\
 & \ge  \frac{1}{2}\|u\|^{2}-\big(\frac{c_2}{\varrho}+1\big)
 \|u\|^{\varrho}_{\varrho}-  \frac{c_2}{p}\|u\|^{p}_{p} \\
 & \ge  \frac{1}{2}\|u\|^{2}- \big(\frac{c_2}{\varrho}+1\big)
 \gamma^{\varrho}_{\varrho}\|u\|^{\varrho}- \frac{c_2}{p}\gamma^{p}_{p}\|u\|^{p}.
\end{aligned}
 \end{equation}
By \eqref{ce} and $2<\varrho\le p$, for $\|u\|$ small there exists a 
$\hat{\kappa}>0$ independent of $\epsilon$ such that 
$\hat{c}_{\epsilon}\ge \hat{\kappa}_{\epsilon}$.
 \end{proof}

 
\begin{proof}[Proof of Theorem \ref{thm1.3}]
  Under the conditions of Theorem \ref{thm1.3}-(a), it is not difficult to verify that 
$F_{\epsilon}$  satisfies (H2), (H3) and (H7). 
In view of Theorem \ref{thm4.1}-(I) and Lemma \ref{lem4.3}, for any $\epsilon>0$, 
there exists a sequence  $\{u_n^{\epsilon}\}$ satisfying
 \begin{equation}\label{T01}
 u_n^{\epsilon}\in \mathcal{N}_{\epsilon}^{-}, \quad
 \Phi_{\epsilon}(u_n^{\epsilon})\to c_{\epsilon},\quad
 \| \Phi_{\epsilon}'(u_n^{\epsilon})\|_{E^*}\to 0.
 \end{equation}
 Lemma \ref{lem5.1} yields that $c_{\epsilon}\ge\kappa_0>0$. 
Let $\{\epsilon_n\}$ be a sequence such that $\epsilon_n\searrow 0$ as 
$n\to\infty$, and
 \begin{equation}\label{T00}
 u_n^{\epsilon_n}\in \mathcal{N}_{\epsilon_n}^{-}, \quad
 \Phi_{\epsilon_n}(u_n^{\epsilon_n})\to \bar{c}\ge\kappa_0>0, \quad
 \| \Phi_{\epsilon_n}'(u_n^{\epsilon_n})\|_{E^*}\to 0.
 \end{equation}
 For the sake of notational simplicity, write $u_n=u_n^{\epsilon_n}$. 
The rest of the proof is divided into three steps.
\smallskip

\noindent\textbf{Step 1.}  Prove that $\{u_n\}$ is bounded in $E$.
 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
 $$
 \delta:=\limsup_{n\to\infty}\sup_{y\in \mathbb{R}^N}\int_{B(y,1)}|v_n^{+}|^2\mathrm{d}x=0,
 $$
then by Lions's concentration compactness principle, $v_n^{+}\to 0$ in
 $L^{s}(\mathbb{R}^N)$ for $2<s<2^*$. Fix $R>[2(1+\bar{c})]^{1/2}$;
 it follows from (H2) that
 \begin{equation}\label{L61}
 \limsup_{n\to\infty}\int_{\mathbb{R}^N}F_{\epsilon_n}(x, Rv_n^{+})\mathrm{d}x
 \le \lim_{n\to\infty} (c_2+{\epsilon_n})
\left(R^{\varrho}\|v_n^{+}\|_{\varrho}^{\varrho}+R^{p}\|v_n^{+}\|_{p}^{p}\right)=0.
 \end{equation}
Let $t_n=R/\|u_n\|_{*}$. Using \eqref{3402}, \eqref{T00} and \eqref{L61}, one has
 \begin{align*}
 \bar{c}+o(1)
 & =  \Phi_{\epsilon_n}(u_n)\\
 & \ge  \Phi_{\epsilon_n}(t_nu_n^{+})+\frac{t_n^2}{2}\|u_n^{-}\|_*^2
 = \Phi_{\epsilon_n}(Rv_n^{+})+\frac{R^2}{2}\|v_n^{-}\|_*^2\\
 & =  \frac{R^2}{2}\|v_n\|_*^2-\int_{\mathbb{R}^N}F_{\epsilon_n}(x, Rv_n^{+})\mathrm{d}x\\
 & =  \frac{R^2}{2}+o(1) > \bar{c}+1+o(1),
 \end{align*}
 which is a contradiction. Thus $\delta>0$. Passing to a $\mathbb{Z}^N$-transformation 
and using the same argument as  in \cite[Lemma 4.4]{T3}, one can show that 
$\{\|u_n\|_{*}\}$ is bounded. By \eqref{La68}, $\{u_n\}$ is bounded in $E$.
\smallskip

\noindent\textbf{Step 2.} Verify that $\Phi'(v_0)=0$ and $\Phi(v_0)\le \bar{c}$ 
for some $v_0\in E\setminus\{0\}$.
 By \eqref{T00} and the same argument as in the Proof of Theorem \ref{thm1.1},
we can show that there is a $v_0\in E$ with $v^{+}_0\neq 0$ such that 
$\Phi'(v_0)=0$ and $\Phi(v_0)\le \bar{c}$.
 By Lemma \ref{lem3.8}, $v_0$ is a nontrivial solution of problem \eqref{ps}.
\smallskip

\noindent\textbf{Step 3.} Show that $\Phi(v_0)=\bar{c}=c_0$.
 We infer form Step 2 and $v_0\in \mathcal{N}^{-}$ that $\bar{c}\ge\Phi(v_0)\ge c_0$. 
Next, we prove that $\bar{c}\le c_0$.

 Let $\epsilon$ be any positive number. Then there exists a 
$u_{\epsilon}\in \mathcal{N}^{-}$ such that 
$\Phi(u_{\epsilon})<c_0+\epsilon$, and
 \begin{equation}\label{T11}
 \langle \Phi'(u_{\epsilon}), u_{\epsilon}\rangle=0, \quad
 \langle \Phi'(u_{\epsilon}), v\rangle=0, \quad  \forall v\in E^{-}.
 \end{equation}
Applying \cite[Lemma 3.3]{Q1}, there exist $t_n>0$ and $v_n\in E^{-}$ such that
 $t_nu_{\epsilon}+v_n\in \mathcal{N}_{\epsilon_n}^{-}$. Then
 \begin{equation}\label{qT11}
\begin{aligned}
 c_{\epsilon_n}
 & \le  \Phi_{\epsilon_n}(t_nu_{\epsilon}+v_n)
 =\frac{t_n^2}{2}\|u^{+}_{\epsilon}\|^2-I_{\epsilon_n}(t_nu_{\epsilon}+v_n)  \\
 & =  \frac{t_n^2}{2}\|u^{+}_{\epsilon}\|^2-\frac{1}{2}\|t_nu^{-}_{\epsilon}
 + v_n\|_{*}^2
 -\int_{\mathbb{R}^N}F(t_nu_{\epsilon}+ v_n)\mathrm{d}x
 -\epsilon_n \|t_nu_{\epsilon}+ v_n\|^{\varrho}_{\varrho}
\end{aligned}
 \end{equation}
 which, together with (H20), imply that there exists a
 $K_{\epsilon}>0$ such that $0<t_n<K_{\epsilon}$
 and $\| v_n\|<K_{\epsilon}$. 
By Lemma \ref{lem2.2}-(i), $\| v_n\|_{\varrho}\le C_9 K_{\epsilon}$.
 Using \eqref{n335}, \eqref{0340}, \eqref{Phe1}, \eqref{T11} 
and H\"{o}lder inequality, we are led to
 \begin{equation}\label{qqV25}
\begin{aligned}
 &c_0+\epsilon\\
 & >  \Phi(u_{\epsilon})=\Phi_{\epsilon_n}(u_{\epsilon})
 +\epsilon_n\|u_{\epsilon}\|_{\varrho}^{\varrho} \\
 & \ge  \Phi_{\epsilon_n}(t_n u_{\epsilon}+ v_n)
 +\frac{1}{2}\| v_n\|_{*}^2 +\frac{1-t_n^2}{2}
 \langle \Phi_{\epsilon_n}'(u_{\epsilon}),
 u_{\epsilon}\rangle-t_n\langle \Phi_{\epsilon_n}'(u_{\epsilon}), v_n\rangle \\
 & \ge  c_{\epsilon_n}+\frac{1}{2}\| v_n\|_{*}^2
 +\frac{1-t_n^2}{2}\langle \Phi_{\epsilon_n}'(u_{\epsilon}),
 u_{\epsilon}\rangle-t_n\langle \Phi_{\epsilon_n}'(u_{\epsilon}), v_n\rangle \\
 & =  c_{\epsilon_n}+\frac{1}{2}\| v_n\|_{*}^2
 -\frac{1-t_n^2}{2}\epsilon_n\varrho\|u_{\epsilon}\|_{\varrho}^{\varrho}
 +t_n\epsilon_n\varrho(|u_{\epsilon}|^{\varrho-2}u_{\epsilon}, v_n)_2 \\
 & \ge  c_{\epsilon_n}-\frac{\epsilon_n\varrho}{2}
 \|u_{\epsilon}\|_{\varrho}^{\varrho}
 -K_{\epsilon}\epsilon_n\varrho\|u_{\epsilon}\|_{\varrho}^{\varrho-1}
 \| v_n\|_{\varrho} \\
 & \ge  c_{\epsilon_n}-\frac{\epsilon_n\varrho}{2}
 \|u_{\epsilon}\|_{\varrho}^{\varrho}
 - C_9K^2_{\epsilon}\epsilon_n\varrho\|u_{\epsilon}\|_{\varrho}^{\varrho-1},
\end{aligned}
 \end{equation}
 this yields
 \begin{equation}\label{V25}
 \bar{c}=\lim_{n\to\infty}c_{\epsilon_n}\le c_0+\epsilon.
 \end{equation}
 Since $\epsilon>0$ is arbitrary, one has $\bar{c}\le c_0$. 
Thus $c_0=\bar{c}\ge\kappa_0$, i.e.,
 $$
 \Phi(v_0)= c_0=\inf_{ u\in\mathcal{N}^-} \Phi(u).
 $$
 Theorem \ref{thm1.3}-(a) is proved.
\smallskip

Under the assumptions of Theorem \ref{thm1.3}-(b), one can verify that $F_{\epsilon}$
 satisfies (H2), (H8) and (H23). In view of Theorem \ref{thm4.1}-(ii) and 
Lemma \ref{lem4.3}, for any $\epsilon>0$ there exists a sequence
 $\{u_n^{\epsilon}\}$ satisfying
 \begin{equation}\label{qT01}
 u_n^{\epsilon}\in \mathcal{M}_{\epsilon}, \quad 
\Phi_{\epsilon}(u_n^{\epsilon})\to \hat{c}_{\epsilon}, \quad
\| \Phi_{\epsilon}'(u_n^{\epsilon})\|_{E^*}\to 0.
 \end{equation}
 Lemma \ref{lem5.2} yields that $\hat{c}_{\epsilon}\ge\hat{\kappa}_{\epsilon}>0$. 
Let $\{\epsilon_n\}$ be a sequence such that $\epsilon_n\searrow 0$ as 
$n\to\infty$, and
 \begin{equation}\label{qT00}
 u_n^{\epsilon_n}\in \mathcal{M}_{\epsilon_n}, \quad
\Phi_{\epsilon_n}(u_n^{\epsilon_n})\to \tilde{c}\ge0, \quad
\| \Phi_{\epsilon_n}'(u_n^{\epsilon_n})\|_{E^*}\to 0.
 \end{equation}
 For the sake of notational simplicity, write $u_n=u_n^{\epsilon_n}$.
 Similar to the proof of Theorem \ref{thm1.3}-(a), the rest of the proof is also 
divided into three steps.
\smallskip

\noindent\textbf{Step 1.} Prove that $\{u_n\}$ is bounded in $E$.
 Since \eqref{0340} does not hold under (H9), the argument used in the 
Proof of Theorem \ref{thm1.3}-(a) is not applicable. We modify it as follows.

Arguing by contradiction, suppose that $\| u_n\|\to \infty$. \eqref{qT00} 
implies the existence of an $M>0$
 such that $\Phi_{\epsilon_n}(u_n)\ge -M$ for all $n\in \mathbb{N}$. By (H2) and 
\eqref{Phe1}, we have
 \begin{equation}\label{L332}
\begin{aligned}
 -2M & \le  2\Phi_{\epsilon_n}(u_n)= \|u^{+}_n\|^2-\|u^{-}_n\|_*^2
 -2\int_{\mathbb{R}^N}F_{\epsilon_n}(x, u_n)\mathrm{d}x \\
 & \le  \|u^{+}_n\|^2-\|u^{-}_n\|_*^2-\frac{2c_1}{\varrho}
\Big(\int_{|u_n|<1}|u_n|^{\varrho}\mathrm{d}x
 +\int_{|u_n|\ge 1}|u_n|^2\mathrm{d}x\Big).
\end{aligned}
 \end{equation}
 From \eqref{315}, \eqref{316} and \eqref{L332}, one sees that
 \begin{equation}\label{Lq334}
\begin{aligned}
 \|u^{-}_n\|_{\varrho}^{\varrho}
 & \le  C_{10}\Big[\|u^{-}_n\|_*^{\varrho}+\int_{|u_n|<1}|u^{-}_n|^{\varrho}
 \mathrm{d}x
 +\Big(\int_{|u_n|\ge 1}|u^{-}_n|^2\mathrm{d}x\Big)^{\varrho/2}\Big] \\
 & \le  C_{10}\|u^{-}_n\|_*^{\varrho}+C_{11}
\Big(\int_{|u_n|<1}|u^{+}_n|^{\varrho}\mathrm{d}x
 +\int_{|u_n|<1}|u_n|^{\varrho}\mathrm{d}x\Big) \\
 & \quad +C_{11}\Big(\int_{|u_n|\ge 1}|u^{+}_n|^2\mathrm{d}x
 +\int_{|u_n|\ge 1}|u_n|^2\mathrm{d}x\Big)^{\varrho/2} \\
 & \le  C_{10}\|u^{-}_n\|_*^{\varrho}+C_{12}
 \Big(\|u^{+}_n\|^{\varrho}+\|u^{+}_n\|^2-\|u^{-}_n\|_*^2+2M\Big) \\
 &\quad  +C_{13}\left(\|u^{+}_n\|^2-\|u^{-}_n\|_*^2+2M\right)^{\varrho/2} \\
 & \le  C_{14}(1+\|u^{+}_n\|^{\varrho}+\|u^{+}_n\|^2),
\end{aligned}
 \end{equation}
 this with \eqref{313}, \eqref{314} and \eqref{L332} imply that
 \begin{align*}
 \|u_n\|^2 & =\|u^{+}_n\|^2+\|u^{-}_n\|_*^2+\|u^{-}_n\|_{\varrho}^2 \\
&\le 2\|u^{+}_n\|^2 +2M
+C_{14}\left(1+\|u^{+}_n\|^{\varrho}+\|u^{+}_n\|^2\right)^{2/\varrho}.
 \end{align*}
 Thus
 \begin{equation}\label{qqn01}
 \|u^{+}_n\|\to \infty \text{ as } n\to \infty, \quad\text{and}\quad
\|u_n\|\le C_{15}(1+\|u^{+}_n\|) .
 \end{equation}
By (H8) and \eqref{Phe1}--\eqref{Ieu}, one has
 \begin{equation}\label{qq01}
\begin{aligned}
 &\Phi_{\epsilon_n}(u_n)-\frac{1}{2} \langle\Phi'_{\epsilon_n}(u_n),u_n\rangle \\
 & =  \int_{\mathbb{R}^N}\big[\frac{1}{2}f(x,u_n)u_n-F(x,u_n)\big]\,\mathrm{d}x
 +\left(\frac{\varrho}{2}-1\right)\epsilon_n\|u_n\|^{\varrho}_{\varrho}  \\
 & \geq  \frac{\mu-2}{2\mu}\int_{\mathbb{R}^N}f(x,u_n)u_n\,\mathrm{d}x
 +\left(\frac{\varrho}{2}-1\right)\epsilon_n\|u_n\|^{\varrho}_{\varrho};
\end{aligned}
 \end{equation}
 then setting $\theta_n=\|\Phi'_{\epsilon_n}(u_n)\|_{E^*}$,
 \begin{equation}\label{qq02}
 \epsilon_n\|u_n\|^{\varrho}_{\varrho} \le C_{16}(1+\theta_n\|u_n\|), 
\quad\text{and}\quad \theta_n\to 0 ~\text{as}~n\to \infty.
 \end{equation}
 Let $\psi\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that $0\le\psi(t)\le 1$ and 
$\psi(t)=0$ if $|t|\le 1$,  $\psi(t)=1$ if $|t|\ge 2$. Set
 \begin{equation}\label{qq03}
 f_1(x,t):=\psi(t) f(x,t), \quad\text{and}\quad 
f_2(x,t):=f(x,t)-f_1(x,t)=(1-\psi(t))f(x,t).
 \end{equation}
Then by (H2) we obtain with $\varrho'=\frac{\varrho}{\varrho-1}$, 
$p'=\frac{p}{p-1}$
 \begin{equation}\label{qq04}
 C_{17}|f_1(x,t)|^{p'}\le tf_1(x,t), \quad
 C_{17}|f_2(x,t)|^{\varrho'}\le tf_2(x,t).
 \end{equation}
 Using \eqref{qT00} and \eqref{qq01}, we see that
 \begin{align*} % \label{qq05}
 \|f_1(x,u_n)\|_{p'}^{p'}+\|f_2(x,u_n)\|_{\varrho'}^{\varrho'}
\le C_{18}(1+\theta_n\|u_n\|).
 \end{align*}
 Moreover, the H\"{o}lder inequality yields
 \begin{equation}\label{qq06}
\begin{aligned}
 \int_{\mathbb{R}^N} f(x,u_n)u^{+}_n \,\mathrm{d}x
 & \le  \|f_1(x,t)\|_{p'}\|u^{+}_n\|_{p}
 +\|f_2(x,t)\|_{\varrho'} \|u^{+}_n\|_{\varrho} \\
 & \le  C_{19}(1+\theta_n\|u_n\|)^{1/p'}\|u^{+}_n\|_{p}
 +C_{19}(1+\theta_n\|u_n\|)^{1/\varrho'}\|u^{+}_n\|_{\varrho}.
\end{aligned}
 \end{equation}
 Then we deduce from \eqref{315}, \eqref{qT00}, \eqref{qqn01}, \eqref{qq02}
 and $\epsilon_n\to 0$ as $n\to \infty$ that
 \begin{equation}\label{qq07}
\begin{aligned}
 \|u^{+}_n\|^2
 & = \langle\Phi'_{\epsilon_n}(u_n),u^{+}_n\rangle 
 +\int_{\mathbb{R}^N} f(x,u_n)u^{+}_n\,\mathrm{d}x
 +\epsilon_n\varrho(|u_n|^{\varrho-2}u_n, u^{+}_n)_{L^2} \\
 & \le  C_{20} \Big( 1+\|u_n\|^{1/p'}+\|u_n\|^{1/\varrho'} \Big)\|u^{+}_n\|
 +\varrho \epsilon_{n}^{1/{\varrho}}
\Big(\epsilon^{1/{\varrho}}_n\|u_n\|_{\varrho}\Big)^{\varrho-1}\|u^{+}_n\|_{\varrho} \\
 & \le  C_{21} \Big( 1+\|u_n\|^{1/p'}+\|u_n\|^{1/\varrho'} \Big)\|u^{+}_n\| \\
 & \le  C_{22} \Big( 1+\|u^{+}_n\|^{1/p'}+\|u^{+}_n\|^{1/\varrho'} \Big)\|u^{+}_n\|.
\end{aligned}
 \end{equation}
 This is a contradiction since $1/\varrho'<1$ and $1/p'<1$.
 Thus $\{\|u_n\|\}$ is bounded.
\smallskip

\noindent\textbf{Step 2.} Verify that $\Phi'(\tilde{v})=0$ and 
$\Phi(\tilde{v})\le \tilde{c}$ for some $\tilde{v}\in E\setminus\{0\}$.
 By \eqref{qT00} and the same argument as in the Proof of Theorem \ref{thm1.1},
 we can certify that there is $\tilde{v}\in E$ with $\tilde{v}^{+}\neq 0$ such 
that $\Phi'(\tilde{v})=0$  and $\Phi(\tilde{v})\le \tilde{c}$. 
This and Lemma \ref{lem3.8},
 imply that $\tilde{v}$  is a nontrivial solution of problem \eqref{ps}.
\smallskip

\noindent\textbf{Step 3.} Show that $\Phi(\tilde{v})=\hat{c}_0=c_0$ if (H5) 
is additionally satisfied.
 Note that if (H23) is satisfied, it does not need to consider the perturbation 
term $\epsilon|u|^{\varrho}$,  and $\Phi_{\epsilon}, \hat{c}_{\epsilon}$ 
in \eqref{qT01} should be replaced by $\Phi$ and $\hat{c}_0$, respectively.
 Then by Step 2 there is a $\tilde{v}\in E$ with $\tilde{v}^{+}\neq 0$ such that
 $\Phi'(\tilde{v})=0$ and $\Phi(\tilde{v})\le \hat{c}_0$. 
This yields that $c_0\le \Phi(\tilde{v})\le \hat{c}_0$
 since $\tilde{v}\in \mathcal{N}^{-}$. 
It follows from \eqref{n17+} that $\hat{c}_0\le c_0$.
 Thus $\Phi(\tilde{v})= \hat{c}_0=c_0\ge \kappa_0>0$. 
In view of \eqref{nq6}, (H23) holds under (H2) and (H5). This completes the proof.
\end{proof}

\section{Appendix}

 Here we show statements (i)--(v) in the proof of Theorem \ref{thm3.1}.

 (i) Suppose that $u_n\in(E_0^{+}\oplus E^{-})\setminus E^{-}$,
 $u_n\to u\in (E_0^{+}\oplus E^{-})\setminus E^{-}$. By (H17), 
$\left\{\hat{n}(u_n)\right\}$ is bounded.
 It suffices to show that $\hat{n}(u_n)\to \hat{n}(u)$ after passing to a 
subsequence. Write $\hat{n}(u_n)=t_nu_n+v_n$ with $v_n\in E^{-}$.
 Passing to a subsequence, we may assume that $t_n\to t$ and 
$v_n\rightharpoonup v$  in $E^{-}$. Then $\hat{n}(u_n)\xrightarrow{\tau} tu+v$. 
Setting $\hat{n}(u)=\bar{t}u+\xi$,
 it follows from (H16) that
 $$
 \Phi(\hat{n}(u_n))
 \ge \Phi\big(\bar{t}u_n+\xi\big)\to \Phi\big(\bar{t}u+\xi\big)=\Phi(\hat{n}(u)),
 $$
 and hence, using the $\tau$-sequentially lower semicontinuity of $I$,
 \begin{align*}
 \Phi(\hat{n}(u))
 & \le  \limsup_{n\to\infty}\Phi(\hat{n}(u_n))\\
& =  \limsup_{n\to\infty}\Big(\frac{1}{2}t^2_n\|u_n^+\|^2-I
\left(\hat{n}(u_n)\right)\Big)\\
 & \le  \frac{1}{2}t^2\|u^+\|^2-I\big(tu+v\big)\\
 &= \Phi(tu+v)\le\Phi(\hat{n}(u)).
 \end{align*}
 Hence the inequalities above must be equalities. 
It follows that $\Phi(tu+v)=\Phi(\hat{n}(u))$
 and $I\left(\hat{n}(u_n)\right)\to I(tu+v)$. 
By (H14), $\hat{n}(u_n) \to tu+v$ and
 so $v_n\to v$. On the other hand, the uniqueness property (H16) yields that
 $\hat{n}(u)=tu+v$. Thus $\hat{n}(u_n)\to \hat{n}(u)$.

 Define $\check{n}: \mathcal{N}^{-}\to S^+$, $z\mapsto z^+/{\|z^{+}\|}$,
 we first certify that $\check{n}(z)\in E_0^{+}$.
 Since $z\in L^{2}(\mathbb{R}^N)$, $\operatorname{meas} \{x\in \mathbb{R}^N: |z(x)|\le \alpha_0\}
=\infty$, it follows from \eqref{a0} that
 \begin{equation}\label{u+}
 \int_{\mathbb{R}^N}\frac{f(x,z) f_{\infty}(x,z)}{V_{\infty}(x)}<0.
 \end{equation}
 By the same argument as in \cite[Lemma 3.2]{Q5}, for any $u\in E$ it holds
 \begin{equation}\label{L262}
\begin{aligned}
 \langle \Phi'(u), u+2v\rangle
 &\ge \|u^{+}\|^2-\| u^{-}+v\|_{*}^2
 +\|v\|_{*}^2-\int_{\mathbb{R}^N}V_{\infty}(x)\left|u+v\right|^2\mathrm{d}x \\
 &\quad  +\int_{\mathbb{R}^N}\frac{V_{\infty}(x) f(x,u) u-|f(x, u)|^2}{V_{\infty}(x)}
 \mathrm{d}x,\quad \forall v\in E^{-}.
\end{aligned}
 \end{equation}
 From \eqref{u+}, \eqref{L262} and the fact $z\in \mathcal{N}^{-}$,
 we deduce for any $v\in E^{-}$ that
 \begin{equation}\label{1203}
\begin{aligned}
 &  \|\check{n}(z)\| -\|v\|_{*}^2
 -\int_{\mathbb{R}^N}V_{\infty}(x)|\check{n}(z)+v|^2\mathrm{d}x \\
 & =  \frac{\|z^{+}\|^2}{\|z^{+}\|^2} -\|v\|_{*}^2
 -\int_{\mathbb{R}^N}V_{\infty}(x)\big|\frac{z}{\|z^+\|}
 -\frac{z^{-}}{\|z^+\|}+v  \big|^2\mathrm{d}x \\
 & \le  \frac{1}{\|z^+\|^2}\langle \Phi'(z), z+2(\|z^+\|v-z^{-})\rangle
 -\frac{1}{\|z^+\|^2}\|{\|z^+\|}v-z^{-}\|_{*}^2 \\
 &\quad  -\frac{1}{\|z^+\|^2}\int_{\mathbb{R}^N}\frac{V_{\infty}(x) f(x,z) \cdot z
 -|f(x,z)|^2}{V_{\infty}(x)}\mathrm{d}x \\
 & =  -\|v-\frac{1}{\|z^+\|}z^{-}\|_{*}^2
 +\frac{1}{\|z^+\|^2}\int_{\mathbb{R}^N}\frac{f(x,z) f_{\infty}(x,z)}{V_{\infty}(x)}<0.
\end{aligned}
 \end{equation}
 Thus $\check{n}(z)\in E_0^+$. It is easy to see that $\check{n}$ 
is continuous and $\check{n}=n^{-1}$
 (the inverse of $n$). Then (i) follows immediately.
 
(ii) Let $u\in E_0^{+}, \ \phi\in E^+$ and put $\hat{n}(u)=s_uu+\xi_u$,
 $\xi_u\in E^{-}$. We claim that $u+t\phi\in E_0^{+}$ for $|t|$ small.
 Arguing indirectly, assume that there exists a sequence $\{t_n\}$ such that
 $u+t_n\phi\not\in E_0^{+}$ and $t_n\to0$. Then there is $\{v_n\}\subset E^{-}$ 
such that
 \begin{equation}\label{Iu+}
 \|u+t_n\phi\|^2-\|v_n\|_{*}^2-\int_{\mathbb{R}^N}V_{\infty}(x)
|u+t_n\phi+v_n|^2\mathrm{d}x\ge0,
 \end{equation}
 this  with \eqref{316} imply that $\{v_n\}$ is bounded in $E$. 
Passing to a subsequence, we may assume that
 $v_n\rightharpoonup v_0$ in $E^{-}$. Then it follows from \eqref{Iu+}
 and the weakly  sequentially lower semicontinuity of the norm that
 \begin{align*}
 \|u\|^2
&= \lim_{n\to\infty}\|u+t_n\phi\|^2\\
 &\ge \liminf_{n\to\infty}\left[\|v_n\|_{*}^2
 +\int_{\mathbb{R}^N}V_{\infty}(x)|u+t_n\phi+v_n|^2\mathrm{d}x\right]\\
 &\ge \|v_0\|_{*}^2+\int_{\mathbb{R}^N}V_{\infty}(x)|u+v_0|^2\mathrm{d}x,
 \end{align*}
which contradicts with the fact that $u\in E_0^{+}$. 
Thus $u+t\phi\in E_0^{+}$ for $|t|$ small.
 
By (H16) and the mean value theorem, we obtain
 \begin{align*}
 \Phi( \hat{n}\left(u+t\phi\right))-\Phi( \hat{n}(u))
 &=  \Phi\left(s_{u+t\phi}(u+t\phi)+\xi_{u+t\phi}\right)-\Phi\left(s_{u}u+\xi_u\right)\\
 &\le \Phi\left(s_{u+t\phi}(u+t\phi)+\xi_{u+t\phi}\right)
 -\Phi\left(s_{u+t\phi}u+\xi_{u+t\phi}\right)\\
 &=  \langle \Phi'\big(s_{u+t\phi}u+\xi_{u+t\phi}
 +t\tau_{t}s_{u+t\phi}\phi\big),ts_{u+t\phi}\phi\rangle
 \end{align*}
 for some $\tau_{t}\in (0,1)$. Similarly,
 \begin{align*}
 \Phi( \hat{n}\left(u+t\phi\right))-\Phi(\hat{n}(u))
 &\ge \Phi\left(s_{u}(u+t\phi)+\xi_{u}\right)-\Phi\left(s_{u}u+\xi_{u}\right)\\
 &=  \langle \Phi'\left(s_{u}u+\xi_{u}+t\eta_{t}s_{u}\phi\right),ts_{u}\phi\rangle
 \end{align*}
 for some $\eta_{t}\in (0,1)$. Since the map $u\mapsto \hat{n}(u)$ 
is continuous according  to (i), we see combining two inequalities that
 \begin{align*}
 \langle(\Phi\circ \hat{n}) '(u),\phi\rangle
 &= \lim_{t\to0}\frac{\Phi( \hat{n}\left(u+t\phi\right))-\Phi( \hat{n}(u))}{t}
 = s_{u}\langle \Phi'\left(s_{u}u+\xi_{u}\right),\phi\rangle\\
 &= \frac{\|\hat{n}(u)^+\|}{\|u\|}\langle \Phi'(\hat{n}(u),\phi\rangle
 \end{align*}
 Hence the G\^{a}teaux derivative of $\Phi\circ \hat{n} $ is bounded linear 
in $\phi$ and continuous  in $u$. It follows that $\Phi\circ \hat{n} $ 
is of class ${C}^1$, see e.g. \cite[Proposition 1.3]{W}.
 Then (ii) holds. (iii), (iv) and (v) are easy consequences of
 (ii) and the definition of $\hat{n}$.

\subsection*{Acknowledgments}
 This work is partially supported by the China Scholarship Council,
 NSFC (Nos. 11571370, 11501190, 11601525, 11626202), 
Natural Science Foundation of Hunan Province, China (No. 2016JJ6137)
 and Research Foundation of Education Bureau of Hunan Province for 
Outstanding Young Teachers (No. 15B223).

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