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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 150, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/150\hfil Schr\"{o}dinger-Maxwell systems]
{Multiple solutions for Schr\"{o}dinger-Maxwell systems with
unbounded and decaying radial potentials}

\author[F. Liao, X. Wang, Z. Liu \hfil EJDE-2014/150\hfilneg]
{Fangfang Liao, Xiaoping Wang, Zhigang Liu}  % in alphabetical order

\address{Fangfang Liao \newline
School of Mathematics and Statistics, Central South University,
Changsha, 410083, Hunan, China.\newline
Department of Mathematics, Xiangnan University,
Chenzhou, 423000, Hunan, China}
\email{liaofangfang1981@126.com}

\address{Xiaoping Wang \newline
Department of Mathematics, Xiangnan University,
Chenzhou, 423000, Hunan, China}
\email{wxp31415@163.com}

\address{Zhigang Liu \newline
Department of Mathematics, Xiangnan University,
Chenzhou, 423000, Hunan, China}
\email{liuzg22@sina.com}

\thanks{Submitted March 14, 2014. Published June 27, 2014.}
\subjclass[2000]{35J20, 35J60}
\keywords{Schr\"{o}dinger-Maxwell system; unbounded or  decaying potential; 
\hfill\break\indent  weighted Sobolev space; mountain pass theorem}

\begin{abstract}
 This article concerns the nonlinear Schr\"{o}dinger-Maxwell system
 \begin{gather*}
 -\Delta u +V(|x|)u +Q(|x|)\phi u=Q(|x|) f(u),\quad \text{in } \mathbb{R}^3\\
 -\Delta \phi =Q(|x|) u^{2}, \quad \text{in } \mathbb{R}^3
 \end{gather*}
 where $V$ and $Q$ are unbounded and decaying radial. Under suitable
 assumptions on nonlinearity $f(u)$, we establish the existence of
 nontrivial solutions and a sequence of high energy solutions in
 weighted Sobolev space via Mountain Pass Theorem and symmetric
 Mountain Pass Theorem.
\end{abstract}

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

\section{Introduction}

This article concerns the  nonlinear Schr\"{o}dinger-Maxwell system
\begin{equation}\label{1.1}
\begin{gathered}
-\Delta u +V(|x|)u +Q(|x|)\phi u=Q(|x|) f(u),\quad \text{in } \mathbb{R}^3\\
-\Delta \phi =Q(|x|) u^{2},\quad\text{in } \mathbb{R}^3.
\end{gathered}
\end{equation}
Such a system, also known as the nonlinear Schr\"{o}dinger-Maxwell
system, arises in an interesting physical context. Indeed, according
to a classical model, the interaction of a charge particle with an
electromagnetic field can be described by coupling the nonlinear
Schr\"{o}dinger and the Maxwell equations. For more details on the
physical aspects, we refer to \cite{BF1}. In particular, if we are
looking for electrostatic-type solutions, we just have to solve
\eqref{1.1}.

For this problem  in a bounded domain, there are some works.
Let us recall some recent results. Benci and Fortunato obtained the
existence of infinitely many solutions of an eigenvalue problem in
\cite{BF1}.  D'Aprile and Wei \cite{DW} studied concentration
phenomena for the system in the unit ball $B_1$ of
$\mathbb{R}^3$ with Dirichlet boundary conditions. Candela and
Salvatore  \cite{CS} considered the problem with a non-homogeneous
term and obtained infinitely many radially symmetric solutions.

Recently, the problem in the whole space $\mathbb{R}^3$ was
considered in some works, see for instance $[4-15]$ 
and the references therein. We recall some of them as follows.
Ruiz \cite{R} considered the  system
\begin{equation}\label{1.2}
\begin{gathered}
-\Delta u +V(x)u +\lambda\phi u=Q(x) f(u),\quad\text{in }\mathbb{R}^3\\
-\Delta \phi = u^{2},\quad\text{in }\mathbb{R}^3
\end{gathered}
\end{equation}
and obtained the existence and nonexistence of radial solutions for
\eqref{1.2} with $V(x)=Q(x)=1,f(u)=u^{p}(1<p<5)$. Later,
Ambrosetti and Ruiz in \cite{AR} obtained multiplicity results for
\eqref{1.2} with $V(x) = Q(x) = 1$. For the critical growth case,
we refer to \cite{ZZ2}. Zhao and Zhao established the existence of a
positive solution by the concentration compactness principle. Sun,
Chen and Nieto \cite{SCN} obtained the existence of ground state
solutions when $V(x)=1, \lambda=K(x)$ and $f(u)$ is asymptotically
linear at infinity. When the potential $V$ is not a constant, Wang
and Zhou \cite{WZ} also considered that the case $f(x,u)$ is
asymptotically linear and the positive potential $V$ is bounded and
non-radial. Mercuri \cite{M} considered the potential $V$ may vanish
at infinity and bounded; i.e., 
$\frac{a}{1+|x|^{\alpha}}\leq V(x)\leq A$ for some $\alpha\in(0,2]$, 
$a, A>0$. By using the
classical Mountain Pass Theorem, the author obtained the existence
of positive solutions with $\lambda=1$ and
$f(u)=u^{p}(1<p<\frac{N+2}{N-2}, N=3,4,5)$. Soon after, Sun, Chen
and Yang \cite{SCY} considered the asymptotically linear case under
the assumptions in \cite{M}, the existence and nonexistence of
solutions are obtained depending on the parameters $\lambda$. When
$\lambda=1$ and $Q=1$, Chen and Tang \cite{CT} considered the
potential $V(x)$ satisfies some coercive condition, i.e.,
\begin{itemize}
\item[(V0)]  $V\in C(\mathbb{R}^3,\mathbb{R})$ satisfies 
$\inf_{x\in\mathbb{R}^3}V(x)>0$ and
for each $M>0$, $\operatorname{meas}\{x\in \mathbb{R}^3|V(x)\le M\}<+\infty$,
\end{itemize}
and proved that \eqref{1.2} has infinitely many high energy
solutions under the condition that $f(x,u)$ is superlinear at
infinity in $u$  by fountain theorem established in \cite{Z}. Soon
after, Li, Su and Wei \cite{LSW} improved their results. For $V(x)$
and $f(x,u)$ are $1$-periodic in each $x$. Zhao and Zhao \cite{ZZ1}
considered this case and obtained the existence of infinitely many
geometrically distinct solutions. For the result of semiclassical
solutions, we refer to \cite{YSD}.

In the present paper, we will consider more general radial
potential, that is, the potential $V(x)$ may be unbounded, decaying
and vanishing. We make the following assumptions:
\begin{itemize}
\item[(V1)]  $V(r)\in C((0,+\infty))$, $V(r)\geq 0$ and there
exist $a_{0}$ and $a_1$ such that
\[
 \liminf _{r\to 0}\frac{V(r)}{r^{a_{0}}}>0,
\quad \liminf _{r\to+\infty}\frac{V(r)}{r^{a_1}}>0,
\]
\item[(Q0)]   $Q(r)\in C((0,+\infty))$, $Q(r)\geq 0$ and there
exist $b_{0}$ and $b_1$ such that
\[
 \limsup _{r\to 0}\frac{Q(r)}{r^{b_{0}}}>0,\quad 
\limsup _{r\to+\infty}\frac{Q(r)}{r^{b_1}}>0.
\]
\end{itemize}
Next we introduce notation. Let $C_{0}^{\infty}(\mathbb{R}^3)$
 denote the collection of smooth functions with compact support and
\[
C_{0,r}^{\infty}(\mathbb{R}^3)=\{u\in C_{0}^{\infty}(\mathbb{R}^3):
u\text{ is radial}\}.
\]
Let $D_{r}^{1,2}(\mathbb{R}^3)$ be the completion of
$C_{0,r}^{\infty}(\mathbb{R}^3)$ under the norm
\[
\|u\|_{D_{r}^{1,2}}=\Big(\int_{\mathbb{R}^3}|\nabla
u|^{2}dx\Big)^{1/2}.
\]
Define
\[
L^{p}(\mathbb{R}^3; Q)=\{u: \mathbb{R}^3\to  \mathbb{R};
u\text{ is measurable and } \int_{\mathbb{R}^3}Q(|x|)|u|^{p}dx<
\infty\}.
\]
with norm
\[
\|u\|_{p}=\Big(\int_{\mathbb{R}^3}Q(|x|)|u|^{p}dx\Big)^{1/p}.
\]
Set
\[
E=H_{r}^{1}(\mathbb{R}^3; V)=D_{r}^{1,2}(\mathbb{R}^3)\cap
L^{2}(\mathbb{R}^3; V),
\]
which is a Hilbert space with the norm
\[
\|u\|_{E}=\Big(\int_{\mathbb{R}^3}|\nabla u|^{2}+V(|x|)u^{2}dx\Big)^{1/2}.
\]
Corresponding to \cite{SWM}, if (V1) and (Q0) are satisfied,
for $N=3$, we define
\begin{gather*}
\overline{p}(a_{0},b_{0})=\begin{cases} 
6+2b_{0}, & b_{0}\geq -2, \; a_{0}\geq -2,\\
\frac{8+4b_{0}-2a_{0}}{4+a_{0}}, & -2\geq a_{0}>-4, \; b_{0}\geq a_{0},\\
\infty , & a_{0}\leq -4, \; b_{0}>-4,
\end{cases}
\\
\underline{p}(a_1,b_1)=\begin{cases}
\frac{8+4b_1-2a_1}{4+a_1}, &b_1\geq a_1>-2,\\
6+2b_1, &b_1\geq-2, \;  a_1\leq-2,\\
2 , & b_1\leq max\{a_1, -2\}.
\end{cases}
\end{gather*}
On the other hand, recently, Su, Wang and Willem \cite{SWM} studied
the nonlinear Schr\"{o}dinger equation
\begin{equation}\label{1.3}
\begin{gathered}
-\Delta u +V(|x|)u =Q(|x|)f(u), \quad\text{in } \mathbb{R}^N\\
u(x)\to 0, \quad \text{as } |x|\to \infty.
\end{gathered}
\end{equation}
and assumed the Ambrosetti-Rabinowitz condition holds; i.e., there
exists $\mu>2$ such that
\begin{eqnarray}\nonumber
0<\mu F(u)\leq uf(u), \quad \forall u \in\mathbb{R},
\end{eqnarray}
where $F(u)=\int_{0}^{u}f(s)ds$. They proved the existence of ground
states solutions when $V$ and $Q$ satisfy the assumption (V1) and
(Q0).

Motivated by the above facts, as in \cite{SWM}, the purpose of this
paper is to extend the existence results of problem \eqref{1.3} to
Schr\"{o}dinger-Maxwell system \eqref{1.1}. Moreover, we assume
\begin{itemize}
\item[(Q1)]  $Q\in L^{\frac{6p-12}{5p-12}}(\mathbb{R}^3)$ for all
$p>12/5$.
\end{itemize}
To reduce our statement, we first make the following
assumption on $f$.
\begin{itemize}
\item[(F1)]  $f\in C(\mathbb{R}, \mathbb{R})$, and 
$|f(u)|\leq c(|u|^{p_1-1}+|u|^{p_{2}-1})$
\end{itemize}
for some $\underline{p}<p_1\leq p_{2}<\overline{p}$
($\underline{p},\overline{p}$ will be defined later), where $c$ is a
positive constant.

 Throughout this article we denote by
$c_{i},C_{i}$ various positive constants, $|\cdot|_{p}$ denotes the
usual $L^{p}(\mathbb{R}^3)$-norm, and $\|\cdot\|_{q}$ denotes the
$L^{q}(\mathbb{R}^3,Q)$-norm.

\section{Preliminaries}

To prove our results, we use the following lemma from \cite{SWM}.


 \begin{lemma} \label{lem2.1}
  Assume {\rm (V1)} and {\rm (Q0)} with
$\overline{p}=\overline{p}(a_{0},b_{0})\geq\underline{p}
=\underline{p}(a_1, b_1)$. Then
\[
H_{r}^{1}(\mathbb{R}^3; V)\hookrightarrow L^{p}(\mathbb{R}^3; Q),
\]
for $\underline{p}\leq p\leq \overline{p}$ when 
$\overline{p}< \infty$ and for $\underline{p}\leq p< \overline{p}$ when
$\overline{p}= \infty$. Furthermore, if 
$ b_1\geq \max\{a_1, -2\}$, the embedding is compact for
$\underline{p}< p< \overline{p}$, and if $ b_1< max\{a_1, -2\}$,
the embedding is compact for $2\leq p< \overline{p}$.
\end{lemma}


 \begin{remark} \label{rmk2.1} \rm
In particular, we can take
$\overline{p}=\overline{p}(a_{0},b_{0})\geq\underline{p}
=\max\{\underline{p}(a_1,b_1),12/5\}$ if we take suitable
$a_{0},b_{0}$. Clearly, 
$ \underline{p}=\max\{\underline{p}(a_1,b_1),\frac{12}{5}\}
\geq {\underline{p}(a_1,b_1)}$. Thus, Lemma \ref{lem2.1} holds
for $ \underline{p}$=max$\{\underline{p}(a_1,
b_1),12/5\}$.
\end{remark}

It is well known that system \eqref{1.1} is the Euler-Lagrange
equation of the functional 
$J: E\times D_{r}^{1,2}(\mathbb{R}^3)\to  \mathbb{R}$ defined by
\[
J(u,\phi)=\frac{1}{2}\|u\|_{E}^{2}-\frac{1}{4}\int_{\mathbb{R}^3}|\nabla
\phi|^{2}dx+\frac{1}{2}\int_{\mathbb{R}^3}Q(|x|)\phi
u^{2}dx-\int_{\mathbb{R}^3}Q(|x|)F(u)dx.
\]
For any $u\in E$, consider the linear functional 
$T_{u}: D_{r}^{1,2}(\mathbb{R}^3)\to  \mathbb{R}$ defined as
\[
T_{u}(v)=\int_{\mathbb{R}^3}Q(|x|)u^{2}v dx.
\]
For $\underline{p}<p<\overline{p}$, by (Q1), the H\"{o}lder
inequality and Lemma \ref{lem2.1}, we have
\begin{align*}
&\int_{\mathbb{R}^3}Q(|x|)u^{2}v dx\\
&= \int_{\mathbb{R}^3}Q(|x|)^{\frac{p-2}{p}}Q(|x|)^{2/p}u^{2}v dx\\
&\leq \Big(\int_{\mathbb{R}^3}Q(|x|)^{\frac{p-2}{p}\cdot\frac{6p}{5p-12}}dx
 \Big)^{\frac{5p-12}{6p}}\Big(\int_{\mathbb{R}^3}(Q(|x|)
^{2/p}u^{2})^{p/2}dx\Big)^{2/p}
\Big(\int_{\mathbb{R}^3}v^{6}dx\Big)^{1/6}\\
&\leq 
S^{-1}|Q|_{\frac{6p-12}{5p-12}}^{\frac{p-2}{p}}
\int_{\mathbb{R}^3}\big(Q(|x|)u^{p}\big)^{2/p}\|v\|_{D^{1,2}_{r}}\\
&\leq
c_1S^{-1}|Q|_{\frac{6p-12}{5p-12}}^{\frac{p-2}{p}}\|u\|_{E}^{2}
\|v\|_{D^{1,2}_{r}}.
\end{align*}
where $S$ is the best Sobolev embedding constant. Hence, the
Lax-Milgram theorem implies that for every $u\in E$, there exists a
unique $\phi _{u}\in D_{r}^{1,2}(\mathbb{R}^3)$ such that
\[
\int_{\mathbb{R}^3}Q(|x|)u^{2}v
= \int_{\mathbb{R}^3}\nabla\phi_{u}\cdot\nabla v, \quad\text{for any }
v\in D_{r}^{1,2}(\mathbb{R}^3),
\]
Using the integration by parts, we obtain
\[
\int_{\mathbb{R}^3}\nabla\phi _{u} \nabla v dx =-\int_{\mathbb{R}^3}
v \Delta \phi _{u} dx, \quad\text{for any } v\in
D_{r}^{1,2}(\mathbb{R}^3);
\]
therefore,
$-\Delta\phi _{u}=Q(|x|)u^{2}$.
 We can write an integral expression for $\phi _{u}$
in the form
\begin{equation}\label{1.4}
\phi
_{u}=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{Q(|y|)u^{2}(y)}{|x-y|}dy,
\end{equation}
for any $u \in C_{0}^{\infty}(\mathbb{R}^3)$, by density it can be
extended for any $u \in E$. Moreover, the functions $\phi _{u}$
possess the following properties:
\[
\phi _{u}\geq 0,\quad \|\phi _{u}\|_{D_{r}^{1,2}}\leq
c_{2}\|u\|_{p}^{2}\leq c_{3}\|u\|_{E}^{2}.
\]
In fact, clearly, $\phi _{u}\geq 0$ by \eqref{1.4}. Using
integration by parts, $-\Delta \phi _{u}=Q(|x|)u^{2}$, the
H\"{o}lder inequality and the Sobolev inequality, for any $u \in E$,
we obtain
\begin{align*}
\|\phi _{u}\|_{D_{r}^{1,2}}^{2}
&= \int_{\mathbb{R}^3} \nabla \phi _{u}\cdot \nabla\phi _{u} dx=-\int_{\mathbb{R}^3} \Delta\phi _{u}\cdot\phi _{u} dx\\
&= \int_{\mathbb{R}^3}Q(|x|) \phi _{u}u^{2} dx\\
&\leq c_1S^{-1}|Q|_{\frac{6p-12}{5p-12}}^{\frac{p-2}{p}}\|u\|_{p}^{2}\|\phi _{u}\|_{D^{1,2}_{r}}\\
&\leq  c_{2}|u|_{p}^{2}\|\phi _{u}\|_{D^{1,2}_{r}}.
\end{align*}
It follows that
\[
\|\phi_{u}\|_{D_{r}^{1,2}}\leq c_{2}\|u\|_{p}^{2}\leq
c_{3}\|u\|_{E}^{2}.
\]
Moreover, there exists $c_4>0$ such that
\begin{equation}\label{1.5}
\int_{\mathbb{R}^3}Q(|x|) \phi _{u}u^{2} dx \leq c_4\|u\|_{E}^{4}.
\end{equation}
So, we can consider the functional $I: E\to  \mathbb{R}^3$
defined by $I(u)=J(u,\phi_{u})$. By \eqref{1.4} the reduced
functional takes the form
\begin{equation}\label{1.6}
I(u)=\frac{1}{2}\|u\|_{E}^{2}+\frac{1}{4}\int_{\mathbb{R}^3}
Q(|x|)\phi _{u}u^{2} dx -\int_{\mathbb{R}^3}Q(|x|)F(u)dx.
\end{equation}
It is clear that $I$ is well defined.  Moreover, Our hypotheses
imply that $I\in C^{1}(E, \mathbb{R})$ and a standard argument shows
that $(u,\phi)\in E\times D_{r}^{1,2}(\mathbb{R}^3)$ is a critical
point of $J$ if and only if $u$ is a critical point of $I$ and
$\phi=\phi_{u}$ (see \cite{W1}).

 \begin{lemma} \label{lem2.2}
 If assumptions {\rm (V1), (Q0), (Q1), (F1)} hold, then 
$I\in C^{1}(E, \mathbb{R})$ and
\begin{equation}\label{2.1}
\langle I'(u), v\rangle=\int_{\mathbb{R}^3}(\nabla u\cdot
\nabla v +V(|x|)uv)dx+\int_{\mathbb{R}^3} Q(|x|)\phi _{u}uv
dx-\langle \Psi'(u), v\rangle,
\end{equation}
where $\Psi(u)=\int_{\mathbb{R}^3}Q(|x|)F(u)dx$.
\end{lemma}

\begin{proof}
First, we prove the existence of the Gateaux derivative
of $\Psi$. From (F1), we have
\begin{gather}\label{2.2}
|f(u)|\leq c(|u|^{p_1-1}+|u|^{p_{2}-1}),\\
\label{2.3}
|F(u)|\leq c(\frac{1}{p_1}|u|^{p_1}+\frac{1}{p_{2}}|u|^{p_{2}}).
\end{gather}
For any $u, v \in E$ and $0<|t|<1$, by the mean value and
\eqref{2.2}, there exists $0<\theta<1$ such that
\begin{align*}
&\frac{|Q(|x|)F(u+tv)-Q(|x|)F(u)|}{|t|}\\
&= |Q(|x|)f(u+\theta tv)v|\\
&\leq  cQ(|x|)(|u+\theta tv|^{p_1-1}+|u+\theta tv|^{p_{2}-1})|v|\\
&\leq
c_{5}Q(|x|)[(|u|^{p_1-1}|v|+|v|^{p_1})+(|u|^{p_{2}-1}|v|+|v|^{p_{2}})]
\end{align*}
The H\"{o}lder inequality implies 
\[
g(x):=
c_{}Q(|x|)[(|u|^{p_1-1}|v|+|v|^{p_1})+(|u|^{p_{2}-1}|v|+|v|^{p_{2}})]\in
L^{1}(\mathbb{R}^3).
\]
Consequently, by the Lebesgue's dominated convergence theorem, one
has
\[
\langle \Psi'(u), v\rangle=\int_{\mathbb{R}^3}Q(|x|)f(u)vdx.
\]
Next,we show that $\Psi'(\cdot):E\to  E^{\ast}$ is
continuous.
Assume that $u_{n}\to  u$ in $E$. By Lemma \ref{lem2.1}, we know that
$u_{n}\to  u$ in $L^{p}(\mathbb{R}^3; Q)$,
for $\underline{p}\leq p\leq \overline{p}$ when 
$\overline{p}< \infty$ and for $\underline{p}\leq p< \overline{p}$ when
$\overline{p}= \infty$.

 On the space $L^{p_1}(\mathbb{R}^3;Q)\cap L^{p_{2}}(\mathbb{R}^3; Q)$,
 we define the norm
\begin{align*}
\|u\|_{p_1\wedge p_{2}}
&= \|u\|_{{p_1}}+\|u\|_{{p_{2}}}\\
&= \Big(\int_{\mathbb{R}^3}Q(|x|)|u|^{p_1}dx\Big)^{1/p_1}+
\left(\int_{\mathbb{R}^3}Q(|x|)|u|^{p_{2}}dx\right)^{1/p_2}
\end{align*}
On the space $L^{p_1}(\mathbb{R}^3; Q)+ L^{p_{2}}(\mathbb{R}^3;Q)$,
we define the norm
\[
\|u\|_{p_1\vee p_{2}}=\inf \left\{\|v\|_{{p_1}}+\|w\|_{{p_{2}}}:
v\in L^{p_1}(\mathbb{R}^3; Q), w\in L^{p_{2}}(\mathbb{R}^3; Q),
u=v+w\right\}.
\]
Since $\underline{p}<p_1\leq p_{2}<\overline{p}$, one has
$u_{n}\to  u$ in 
$L^{p_1}(\mathbb{R}^3; Q)\cap L^{p_{2}}(\mathbb{R}^3; Q)$.
Similar to \cite[Theorem A.4]{W1}, we  have
\[
f(u_{n})\to  f(u)\quad \text{in }
 L^{p_1'}(\mathbb{R}^3; Q)+ L^{p_{2}'}(\mathbb{R}^3; Q).
\]
By the H\"{o}lder inequality, we have
\begin{align*}
|\langle \Psi'(u_{n})-\Psi'(u), v\rangle| 
&\leq  \|f(u_{n})-f(u)\|_{p_1'\vee p_{2}'}\|v\|_{p_1\wedge p_{2}}\\
&\leq  c_{6}\|f(u_{n})- f(u)\|_{p_1'\vee p_{2}'}\|v\|_{E},
\end{align*}
where $p_{i}'=p_{i}/(p_{i}-1)$, $i=1, 2$. Hence
\[
\|\Psi'(u_{n})-\Psi'(u)\|\leq c_{6}\|f(u_{n})- f(u)\|_{p_1'\vee
p_{2}'}\to  0
 \quad\text{as } n\to  \infty.
\]
This shows $\Psi'(\cdot):E\to  E^{\ast}$ is continuous. This
completes the proof.
\end{proof}


 \begin{lemma} \label{lem2.3}
 Under the condition {\rm (F1)}, if $\{u_{n}\}\subset E$ is a bounded 
sequence with $I'(u_{n})\to  0$, then
$\{u_{n}\}$ has a convergent subsequence.
\end{lemma}

\begin{proof} 
Since $\{u_{n}\}\subset E$ is bounded and the embedding
$E \hookrightarrow L^{s}(\mathbb{R}^3;Q)$
 is compact for each $s\in
(\underline{p}, \overline{p})$, passing to a subsequence, we can
assume that
$u_{n}\rightharpoonup u$ in $E$, and
\[
u_{n}\to  u \quad\text{in } L^{s}(\mathbb{R}^3; Q),\; s\in
(\underline{p}, \overline{p}).
\]
Note that
\begin{align*}
& \langle I'(u_{n})-I'(u), u_{n}-u \rangle\\
&= \|u_{n}-u\|_{E}^{2}+\int_{\mathbb{R}^3}
\left(Q(|x|)\phi_{u_{n}}u_{n}^{2}-Q(|x|)\phi_{u_{n}}u_{n}u\right)dx\\
&\quad +\int_{\mathbb{R}^3}
\left(Q(|x|)\phi_{u}u^{2}-Q(|x|)\phi_{u}u_{n}u\right)dx
-\int_{\mathbb{R}^3}Q(|x|)\left(f(u_{n})- f(u)\right)(u_{n}-u)dx.
\end{align*}
We have
\begin{align*}
& \|u_{n}-u\|_{E}^{2}\\
&= \langle I'(u_{n})-I'(u), u_{n}-u \rangle -\int_{\mathbb{R}^3}
\left(Q(|x|)\phi_{u_{n}}u_{n}^{2}-Q(|x|)\phi_{u_{n}}u_{n}u\right)dx\\
&\quad -\int_{\mathbb{R}^3}
\left(Q(|x|)\phi_{u}u^{2}-Q(|x|)\phi_{u}u_{n}u\right)dx
+\int_{\mathbb{R}^3}Q(|x|)\left(f(u_{n})- f(u)\right)(u_{n}-u)dx.
\end{align*}
Since
$u_{n}\rightharpoonup u$ in $E$ 
and $I'(u_{n})\to  0$, we have
\[
\langle I'(u_{n})-I'(u), u_{n}-u \rangle \to  0
 \quad\text{as } n\to \infty.
\]
On one hand, by (Q1), for $\underline{p}<p<\overline{p}$, we
have
\begin{align*}
\int_{\mathbb{R}^3}Q(|x|)
(\phi_{u_{n}}u_{n}^{2}-\phi_{u_{n}}u_{n}u)dx
&= \int_{\mathbb{R}^3}Q(|x|) \phi_{u_{n}}u_{n}(u_{n}-u)dx\\
&\leq  |Q|_{\frac{6p-12}{5p-12}}^{\frac{p-2}{p}}\|u_{n}-u\|_{p}\|u_{n}\|_{p}\|\phi_{u_{n}}\|_{6}\\
&\leq
c_{7}\|u_{n}-u\|_{p}\|u_{n}\|_{p}\|\phi_{u_{n}}\|_{D_{r}^{1,2}}.
\end{align*}
Hence,
\[
\int_{\mathbb{R}^3}Q(|x|)
\left(\phi_{u_{n}}u_{n}^{2}-\phi_{u_{n}}u_{n}u\right)dx \to
0, \quad  \text{as } n\to \infty.
\]
Similarly,
\[
\int_{\mathbb{R}^3}Q(|x|)\left(\phi_{u}u^{2}-\phi_{u}u_{n}u\right)dx \to  0,
 \quad \text{as } n\to \infty.
\]
On the other hand,
\begin{align*}
& |\int_{\mathbb{R}^3}Q(|x|)\left(f(u_{n})-
f(u)\right)(u_{n}-u)dx|\\
& \leq
\int_{\mathbb{R}^3}Q(|x|)\left(|f(u_{n})|+|f(u)|\right)|u_{n}-u|dx\\
&\leq c\int_{\mathbb{R}^3}Q(|x|)\left(|u_{n}|^{p_1-1}+|u_{n}|^{p_{2}-1}+
|u|^{p_1-1}+|u|^{p_{2}-1}\right)|u_{n}-u|dx\\
&\leq c\Big(\int_{\mathbb{R}^3}Q(|x|)|u_{n}-u|^{p_1}dx\Big)^{1/p_1}
\Big(\Big(\int_{\mathbb{R}^3}Q(|x|)|u_{n}|^{p_1}dx\Big)^{\frac{p_1-1}{p_1}}\\
&\quad +\Big(\int_{\mathbb{R}^3}Q(|x|)|u|^{p_1}dx\Big)^{\frac{p_1-1}{p_1}}\Big)\\
&\quad+c\Big(\int_{\mathbb{R}^3}Q(|x|)|u_{n}-u|^{p_{2}}dx\Big)^{1/p_2}
\Big(\Big(\int_{\mathbb{R}^3}Q(|x|)|u_{n}|^{p_{2}}dx\Big)^{\frac{p_{2}-1}{p_{2}}}\\
&\quad +\Big(\int_{\mathbb{R}^3}Q(|x|)|u|^{p_{2}}dx\Big)^{\frac{p_{2}-1}{p_{2}}}\Big).
\end{align*}
Since $u_{n}\to  u$ in $L^{s}(\mathbb{R}^3; Q), s\in(\underline{p}, \overline{p})$, 
we have
\[
\int_{\mathbb{R}^3}Q(|x|)\left(f(u_{n})- f(u)\right)(u_{n}-u)dx
\to  0 \quad \text{as } n\to \infty.
\]
So we have $\|u_{n}-u\|_{E}\to  0$. This completes the proof.
\end{proof}

\section{Main results}

\begin{theorem} \label{thm3.1}
Assume that conditions {\rm (V1), (Q0), (Q1)} hold. 
If (F1) and the following conidition hold
\begin{itemize}
\item[(F2)] There exists $\mu$ and $r>0$  such that 
$\max\{\underline{p},4\}<\mu \leq\overline{p}<\infty$, and 
\[
\mu F(u)\leq uf(u), ~\forall u \in\mathbb{R}, \quad 
 \inf _{|u|= r}F(u):=\beta>0.
\]
\end{itemize}
Then system \eqref{1.1} has a nontrivial solution.
Furthermore, if $f(u)$ is odd in $u$, then system \eqref{1.1}
has a sequence $\{(u_{n},\phi_{n})\}$ of solutions in 
$E\times D_{r}^{1,2}(\mathbb{R}^3)$ with $\|u_{n}\|\to  \infty$ and
$I(u_{n})\to  +\infty$.
\end{theorem}

 \begin{proof} From (F1), we have
\[
|F(u)|\leq c(\frac{1}{p_1}|u|^{p_1}+\frac{1}{p_{2}}|u|^{p_{2}}).
\]
Note that
\begin{align*}
I(u)
&= \frac{1}{2}\|u\|_{E}^{2}+\frac{1}{4}\int_{\mathbb{R}^3} Q(|x|)\phi
_{u}u^{2} dx -\int_{\mathbb{R}^3}Q(|x|)F(u)dx\\
&\geq \frac{1}{2}\|u\|_{E}^{2}-\int_{\mathbb{R}^3}Q(|x|)F(u)dx\\
&\geq \frac{1}{2}\|u\|_{E}^{2}-\frac{c}{p_1}\|u\|_{p_1}^{p_1}
 -\frac{c}{p_{2}}\|u\|_{p_{2}}^{p_{2}}\\
&\geq \frac{1}{2}\|u\|_{E}^{2}-c_{8}\|u\|_{E}^{p_1}-c_{9}\|u\|_{E}^{p_{2}}.
\end{align*}
Since $p_1, p_{2}>2$, we can take a small $\rho$ such that
\[
I|_{\partial B_{\rho}}\geq
\frac{1}{2}\rho^{2}-c_{8}\rho^{p_1}-c_{9}\rho^{p_{2}}:=\delta>0,
\]
where $B_{\rho}=\{u\in E: \|u\|_{E}<\rho\}$.

For $z\in \mathbb{R}$, set
\[
h(t):=F(t^{-1}z)t^{\mu},\quad \forall t\in [1, +\infty).
\]
For $|z|\geq r$ and $t\in [1, |z|/r]$, by (F2), one has
\begin{align*}
h'(t)
&= f(t^{-1}z)(-\frac{z}{t^{2}})t^{\mu}+F(t^{-1}z)\mu t^{\mu-1}\\
&= t^{\mu-1}\left(\mu F(t^{-1}z)-t^{-1}zf(t^{-1}z)\right)
\leq  0.
\end{align*}
So, we have
\[
F(z)=h(1)\geq h(\frac{|z|}{r})\geq\frac{\beta}{r^{\mu}}|z|^{\mu}.
\]
Since $\mu>4$, there exists a constant
$\max\{\underline{p},4\}<\alpha <\overline{p}$ such that
$\alpha<\mu$, and hence
\begin{equation}\label{3.1}
 \lim _{|u|\to \infty}\frac{F(u)}{|u|^{\alpha}}=+\infty.
\end{equation}

For any finite dimensional space $E_1\subset E$, by the
equivalence of norms in the finite space, there exists a constant
$c_{(\alpha)}>0$, such that
\begin{equation}\label{3.2}
\|u\|_{\alpha}\geq c_{\alpha}\|u\|_{E},\quad \forall u\in E_1
\end{equation}
where $\alpha$ is the constant appearing in \eqref{3.1}. For any
$\sigma >0$, by (F1), there is a constant $c_{\sigma}>0$ such
that
\[
|F(u)|\leq c_{\sigma}|u|^{\underline{p}},\quad \forall |u|<\sigma.
\]
Hence, by \eqref{3.1}, we know that for $M>0$, there is a constant
$C_{M}>0$ such that
\begin{equation}\label{3.3}
F(u)\geq M|u|^{\alpha}-C_{M}|u|^{\underline{p}},\ \ \forall u\in
\mathbb{R}.
\end{equation}
By \eqref{3.2} and \eqref{3.3}, we have
\begin{align*}
I(u)
&\leq \frac{1}{2}\|u\|_{E}^{2}+\frac{c_4}{4}\|u\|_{E}^{4}
-M\|u\|_{\alpha}^{\alpha}+C_{M}\|u\|_{\underline{p}}^{\underline{p}}\\
&\leq  \frac{1}{2}\|u\|_{E}^{2}+\frac{c_4}{4}\|u\|_{E}^{4}
-Mc_{\alpha}^{\alpha}\|u\|_{E}^{\alpha}+C_{M}\|u\|_{E}^{\underline{p}},
\end{align*}
for all $u\in E_1$. Consequently, there is a large $r_1>0$ such
that $I<0$ on $E_1\backslash B_{r_1}$. Consequently, there is a
point $e\in E$ with $\|e\|_{E}>\rho$ such that $I(e)<0$.

Now, we prove that $I$ satisfies the Palais-Smale condition. By
Lemma \ref{lem2.3} we know that it is sufficient to prove $\{u_{n}\}$ is
bounded in $E$. Indeed, if a sequence $\{u_{n}\}\subset E$ such that
$I(u_{n})$ is bounded and $I'(u_{n})\to  0$, then there is
positive constant $M_{0}$ such that for large $n$, one has
\begin{align*}
M_{0}+\|u_{n}\|_{E}
&\geq I(u_{n})-\frac{1}{\mu}\langle I'(u_{n}), u_{n} \rangle \\
&\geq (\frac{1}{2}-\frac{1}{\mu})\|u_{n}\|_{E}^{2}+(\frac{1}{4}-\frac{1}{\mu})\int_{\mathbb{R}^3}
Q(|x|)\phi _{u_{n}}u_{n}^{2}dx\\
&\quad +\int_{\mathbb{R}^3}Q(|x|)\Big(\frac{f(u_{n})u_{n}}{\mu}-F(u_{n})\Big)dx\\
&\geq (\frac{1}{2}-\frac{1}{\mu})\|u_{n}\|_{E}^{2}.
\end{align*}
This implies $\{u_{n}\}$ is bounded. 

Obviously, $I(0)=0$. Hence $I$ possesses a critical value 
$\eta\geq \delta$ by \cite[Theorem 2.2]{R1}, thus problem
\eqref{1.1} has a nontrivial solution. Moreover, obviously, $I$ is
bounded on each bounded subset of $E$ and $f(u)$ is odd which
implies $I$ is even. Hence the second conclusion follows from
\cite[Theorem 9.12]{R1}. This completes the proof.
\end{proof}

Note that $\mu>4$ in condition (F2). Now, we consider the weak
case $\mu=4$. At this one, we have the following Theorem.

 \begin{lemma} \label{lem3.2}
 Assume that conditions {\rm (V1), (Q0), (Q1), (F1)} and the following 
conditions hold:
\begin{itemize}
\item[(F3)] $\frac{F(u)}{|u|^{4}}\to  +\infty$ as $|u|\to  +\infty$.
\item[(F4)] $u f(u)\geq 4F(u)$ for all $u \in\mathbb{R}$.
\end{itemize}
If $\underline{p}<4<\overline{p}$, then system \eqref{1.1} has
at least one nontrivial solution. Furthermore, if $f(u)$ is odd in
$u$, then system \eqref{1.1} has a sequence
$\{(u_{n},\phi_{n})\}$ of solutions in $E\times
D_{r}^{1,2}(\mathbb{R}^3)$ with $\|u_{n}\|\to  \infty$ and
$I(u_{n})\to  +\infty$.
\end{lemma}

\begin{proof} 
From the proofs of the first segment in Theorem \ref{thm3.1},
we know that there exist constants $\rho>0$ and $\delta>0$ such that
\[
I|_{\partial B_{\rho}}\geq \delta>0.
\]
Moreover, for any finite dimensional space $E_1\subset E$, by the
equivalence of norms in the finite space, there exists a constant
$C>0$, such that
\begin{equation}\label{3.4}
\|u\|_4\geq C\|u\|_{E},\quad \forall u\in E_1.
\end{equation}
Since $\underline{p}<4$, by (F1) and (F3) we know that for
any $M>\frac{c_4}{4{C}^{4}}$, there is a constant $C_{M}>0$ such
that
\begin{equation}\label{3.5}
F(u)\geq M|u|^{4}-c(M)|u|^{\underline{p}},\quad \forall u\in
\mathbb{R}.
\end{equation}
Hence
\[
I(u)\leq \frac{1}{2}\|u\|_{E}^{2}+\frac{1}{4}\int_{\mathbb{R}^3}
Q(|x|)\phi _{u_{n}}u_{n}^{2} dx
-M\|u\|_4^{4}+C_{M}\|u\|_{\underline{p}}^{\underline{p}}.
\]
By \eqref{3.4} and \eqref{3.5}, we know
\[
I(u)\leq \frac{1}{2}\|u\|_{E}^{2}+\frac{c_4}{4}\|u\|_{E}^{4}
-MC^{4}\|u\|_{E}^{4}+C_{M}\|u\|_{E}^{\underline{p}}\,,
\]
for all $u\in E_1$. Consequently, there is a large $r_1>0$ such
that $I<0$ on $E_1\backslash B_{r_1}$. Consequently, there is a
point $e\in E$ with $\|e\|_{E}>\rho$, such that $I(e)<0$.\par Next
we prove that $I$ satisfies the Palais-Smale condition. Indeed, if a
sequence $\{u_{n}\}\subset E$ is such that $\{I(u_{n})\}$ is bounded
and $I'(u_{n})\to  0$, then there is a positive constant
$M_1$ such that for large $n$, one has
\begin{align*}
M_1+\|u_{n}\|_{E}
&\geq I(u_{n})-\frac{1}{4}\langle I'(u_{n}), u_{n} \rangle \\
&= \frac{1}{4}\|u_{n}\|_{E}^{2}+
\int_{\mathbb{R}^3}Q(|x|)\big(\frac{1}{4}f(u_{n})u_{n}-F(u_{n})\big)dx\\
&\geq \frac{1}{4}\|u_{n}\|_{E}^{2}.
\end{align*}
This implies
$\{u_{n}\}$ is bounded. Hence $\{u_{n}\}\subset E$ has a convergent
subsequence by Lemma \ref{lem2.3}. This shows that $I$ satisfies the
Palais-Smale condition. Finally, the conclusions follows from
\cite[Theorem 2.2 and 9.12]{R1}.
\end{proof}

\begin{corollary} \label{coro3.3}
Assume that conditions {\rm (V1), (Q0), (Q1), (F1), (F3)} and the following
conditions hold:
\begin{itemize}
\item[(F4')]
$u\to f(u)/|u|^3$ is increasing on $(-\infty,0)$
and on $(0,+\infty)$.
\end{itemize}
If $\underline{p}<4<\overline{p}$, then system \eqref{1.1} has
at least one nontrivial solution. Furthermore, if $f(u)$ is odd in
$u$, then system \eqref{1.1} has a sequence
$\{(u_{n},\phi_{n})\}$ of solutions in 
$E\times D_{r}^{1,2}(\mathbb{R}^3)$ with $\|u_{n}\|\to  \infty$ and
$I(u_{n})\to  +\infty$.
\end{corollary}

 \begin{proof}
 It is sufficient to prove that (F4') implies (F4). In fact, whenever $u>0$,
\[
F(u)=\int_{0}^{1}f(ut)u\,dt=\int_{0}^{1}\frac{f(ut)}{(ut)^3}u^{4}t^3dt
\leq\int_{0}^{1}\frac{f(u)}{(u)^3}u^{4}t^3dt
=\frac{1}{4}f(u)u.
\]
Whenever $u<0$,
\[
F(u)=\int_{0}^{1}f(ut)u\,dt=-\int_{0}^{1}\frac{f(ut)}{(-ut)^3}u^{4}t^3dt\leq\int_{0}^{1}\frac{f(u)}{(u)^3}u^{4}t^3dt
=\frac{1}{4}f(u)u.
\]
This shows (F4) holds.
\end{proof}

 \begin{theorem} \label{thm3.3}
 Assume that condition {\rm (V1), (Q1), (F1), (F3)} and the following condition hold:
\begin{itemize}
\item[(F5)]
$F(u)\geq 0$ for all $u\in \mathbb{R}$ and $G(s)\leq G(t)$ whenever
$(s, t)\in \mathbb{R}^{+}\times \mathbb{R}^{+}$ and $s\leq t$, where
$G(u)=f(u)u-4F(u)$.
\end{itemize}
If $\underline{p}<4<\overline{p}$, then system \eqref{1.1} has
at least one nontrivial solution. Furthermore, if $f(u)$ is odd in
$u$, then system \eqref{1.1} has a sequence
$\{(u_{n},\phi_{n})\}$ of solutions in $E\times
D_{r}^{1,2}(\mathbb{R}^3)$ with $\|u_{n}\|\to  \infty$ and
$I(u_{n})\to  +\infty$.
\end{theorem}

\begin{proof} 
Similar to the proof of  Lemma \ref{lem3.2}, we know that there
exist $\rho>0$, $\delta>0$ such that
\[
I|_{\partial B_{\rho}}\geq \delta>0.
\]
Moreover, for any finite dimensional subspace $E_1\subset E$,
there is a large $r_1>0$ such that $I<0$ on $E_1\backslash B_{r_1}$.

Now, we prove that $I$ satisfies the Cerami condition. Indeed, if a
sequence $\{u_{n}\}\subset E$ is such that $\{I(u_{n})\}$ is bounded
and $(1+\|u_{n}\|)I'(u_{n})\to  0$, then  we claim that
$\{u_{n}\}$ is bounded. If this is false, then we can assume
$\|u_{n}\|\to +\infty$. Set
$v_{n}=\frac{u_{n}}{\|u_{n}\|_{E}}$, then $\|v_{n}\|_{E}=1$. By
virtue of Lemma \ref{lem2.1}, passing to a subsequence, we may assume
\begin{gather*}
v_{n}\rightharpoonup v \quad \text{in } E,\\
u_{n}\to  u \quad \text{in } L^{s}(\mathbb{R}^3; Q), s\in
(\underline{p}, \overline{p}).
\end{gather*}
Since $\{I(u_{n})\}$ is bounded, there exists a constant $C_1>0$
such that
\[
\int_{\mathbb{R}^3}\frac{Q(|x|) F(u_{n})}{\|u_{n}\|_{E}^{4}}dx \leq
C_1<\infty.
\]
Set $\Omega=\{x\in \mathbb{R}^3: v(x)\neq 0\}$. Then
$|u_{n}(x)|\to  +\infty$ for a.e. $x\in \Omega$. If 
$\operatorname{meas}(\Omega)>0$, then, by (F4)
\[
\frac{F(u_{n})}{\|u_{n}\|_{E}^{4}}
=\frac{F(u_{n})}{|u_{n}|^{4}} |v_{n}(x)|^{4}\to  \infty,
 \quad\text{as } n\to \infty.
\]
Since $Q(|x|)>0$, using Fatou's lemma, we obtain
\[
\int_{\mathbb{R}^3}\frac{Q(|x|)
F(u_{n})}{\|u_{n}\|_{E}^{4}}dx\to \infty.
\]
A contradiction, so $\operatorname{meas}(\Omega)=0$. Therefore, 
$v(x)=0$ a.e. $x\in \mathbb{R}^3$.
Next, as in \cite{J}, we define
\[
 I(t_{n}u_{n})
 = \max _{t\in[0,1]}I(tu_{n}).
\]
For any $M>0$, set
$\tilde{v}_{n}=\sqrt{4M}\frac{u_{n}}{\|u_{n}\|_{E}}=\sqrt{4M}v_{n}$.
Since $|F(u)|\leq c(\frac{1}{p_1}|u|^{p_1}+\frac{1}{p_{2}}|u|^{p_{2}})$ for
 $u\in \mathbb{R}$,
\begin{align*}
|\int_{\mathbb{R}^3}Q(|x|) F(\tilde{v}_{n})dx|\leq
\frac{c}{p_1}\int_{\mathbb{R}^3}Q(|x|) |\tilde{v}_{n}|^{p_1}dx+
\frac{c}{p_{2}}\int_{\mathbb{R}^3}Q(|x|)
|\tilde{v}_{n}|^{p_{2}}dx\to  0,
\end{align*}
as $n \to  \infty$.
Consequently, for large $n$, one has
\begin{align*}
I(t_{n}u_{n})
&\geq I(\tilde{v}_{n})\\
&\geq \frac{1}{2}\|\tilde{v}_{n}\|_{E}^{2}+\frac{1}{4}\int_{\mathbb{R}^3}
Q(|x|)\phi _{\tilde{v}_{n}}\tilde{v}_{n}^{2}
dx-\int_{\mathbb{R}^3}Q(|x|)F(\tilde{v}_{n})dx\\
&\geq M.
\end{align*} 
This means that
$ \lim _{n\to \infty}I(t_{n}u_{n})=\infty$.
In view of the choice of $t_{n}$ we know that 
$\langle I'(t_{n}u_{n}), t_{n}u_{n} \rangle=0$ or $\to  0$. Hence,
by (F5) and the oddness of $f$, one has
\begin{align*}
&\infty\leftarrow 4I(t_{n}u_{n})-\langle
I'(t_{n}u_{n}), t_{n}u_{n} \rangle\\
&= t_{n}^{2}\int_{\mathbb{R}^3}\left(|\nabla
u_{n}|^{2}+V(|x|)|u_{n}|^{2}\right)dx+
\int_{\mathbb{R}^3}Q(|x|)\left(f(t_{n}u_{n})t_{n}u_{n}-4F(t_{n}u_{n})\right)dx\\
&\leq
\|u_{n}\|_{E}^{2}+\int_{\mathbb{R}^3}Q(|x|)\left(f(u_{n})u_{n}-4F(u_{n})\right)dx\\
&= 4I(u_{n})-\langle I'(u_{n}), u_{n} \rangle.
\end{align*}
This is a contradiction, so $\{u_{n}\}$ is bounded. Consequently,
$\{u_{n}\}\subset E$ has a convergent subsequence by Lemma \ref{lem2.3}.
This shows that $I$ satisfies the Cerami condition. Note that if we
use Cerami condition in place of the Palais-Smale condition, then
\cite[Theorems 2.2 and 9.12]{R1} are still true. Therefore, the
conclusion follows from \cite[Theorems 2.2 and 9.12]{R1}. This
completes the proof.
\end{proof}

\subsection*{Acknowledgments}
This work is partially supported by the NNSF (11171351)
of China, the National Natural Science Foundation of Hunan Province (14JJ2133),the 
Scientific Research Foundation of Hunan Provincial Education Department (13A093),
Scientific Research Fund of Hunan Provincial Education  Department (12C0895),
and the Construct Program of the Key Discipline in Hunan Province.


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\end{document}