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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 276, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/276\hfil Solutions for Schr\"odinger-Poisson systems]
{Multiple solutions for Schr\"odinger-Poisson systems with sign-changing
potential and critical nonlinearity}

\author[L. Shao, H. Chen \hfil EJDE-2016/276\hfilneg]
{Liuyang Shao, Haibo Chen}

\address{Liuyang Shao \newline
 School of Mathematics and Statistics,
 Central South University,
 Changsha, 410083 Hunan, China}
\email{sliuyang316@163.com}

\address{Haibo Chen (corresponding author)\newline
 School of Mathematics and Statistics,
 Central South University,
 Changsha, 410083 Hunan, China}
\email{math\_chb@163.com}

\thanks{Submitted March 12, 2016. Published October 17, 2016.}
\subjclass[2010]{35B38, 35G99}
\keywords{Schr\"odinger-Poisson system; variational methods;
\hfill\break\indent mountain pass theorem; Ekeland's variational principle}

\begin{abstract}
 In this article, we study the Schr\"odinger-Poisson system
 \begin{gather*}
 -\Delta u+V(x)u+k(x)\phi(x)u=h_1(x)|u|^{4}u+\mu h_{2}(x)u+h_3(x),
 \quad\text{in } \mathbb{R}^3, \\
 -\Delta \phi(x)=k(x)u^2 , \quad\text{in } \mathbb{R}^3,
 \end{gather*}
 where $h_1(x),h_{2}(x),h_3(x)$ and $V(x)$ are allowed to be sign-changing
 and $\mu>0$ is a parameter. Under some appropriate assumptions on $V(x)$,
 we obtain the existence of two different solutions for the above system
 via variational methods.
\end{abstract}

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


\section{Introduction and statement of main results}\label{section1}

In this article we consider the  Schr\"odinger-Poisson system
\begin{equation} \label{e1.1}
 \begin{gathered}
 -\Delta u+V(x)u+k(x)\phi(x)u=h_1(x)|u|^{4}u+\mu h_{2}(x)u+h_3(x),
\quad\text{in } \mathbb{R}^3, \\
 -\Delta \phi(x)=k(x)u^2 ,\quad\text{in } \mathbb{R}^3,
 \end{gathered}
 \end{equation}
where $h_1(x),h_{2}(x),h_3(x)$ and $V(x)$ are allowed to be
sign-changing and $\mu>0$ is a parameter. Moreover, $h_3(x)$ is a perturbed term. 
System \eqref{e1.1} is a modified version of the classical 
Schr\"odinger-Poisson system (also called Schr\"odinger-Maxwell equation),
which has a strong physical backgrounds because of its appearances in 
quantum mechanical models and in semiconductor theory. For more details, 
we refer the readers to \cite{b1,c1,l4} and the references therein.

In recent years, with the aid of variational methods, there have been 
many results on existence, nonexistence and multiplicity of solutions for 
such system depending on the assumptions of the potential $V(x)$. 
According to the conditions imposed on the potential $V(x)$, these results 
can be roughly classified into four cases. 
Case 1: Many articles deal with the case when $V(x)$ is a positive constant 
or radially symmetric function, see for example
 \cite{a1,a2,c2,j1,r1,s4,w1}
and the references therein. 
Case 2: There are also a great number of articles devoted to the case when
 $V(x)$ is nonradial, see for instance \cite{j2,l1,l6,s5}.
Case 3: Many articles deal with the case when $V(x)$ possesses some kind 
of periodicity see \cite{l2,s1,s2,s3,s6,t1,t2}.
Case 4: We know that \cite{l3,t1} treat the case when $V(x)$ is sign-changing. 
Here we emphasize problem \eqref{e1.1} not only has sign-changing potential 
$V(x)$ but also possesses critical nonlinearity.

Motivated by the above facts, the goal of this paper is to consider the 
multiplicity of nontrivial solutions for  \eqref{e1.1} when $V(x)$ is 
sign-changing. Under some natural assumptions, by using Mountain Pass Theorem 
in combination with Ekeland's variational principle, the existence results 
of at least two nontrivial solutions are obtained. 
Actually, one positive solution and one negative solution.

Before stating our main results, we give the following assumptions on $V(x)$.
\begin{itemize}
\item[(A1)] $V\in C(\mathbb{R}^3,\mathbb{R})$ and 
$\inf_{x\in \mathbb{R}^3}V(x)>-\infty$.
 Moreover, there exists a constant $d_0$ such that
 $$
\lim_{|y|\to +\infty}\operatorname{meas}\{x\in \mathbb{R}^3:|x-y|
\leq d_0, V(x)\leq M\}=0, \quad \forall M>0.
$$
 where $\operatorname{meas}(\cdot)$ denotes the Lebesgue measure in $\mathbb{R}^3$.
\end{itemize}
 Inspired by Zhang and Xu $[27]$, we can find a constant $V_0>0$ such that
$\tilde{V}(x):=V(x)+V_0\geq a$, where $a>0$ is a constant, and let
$\tilde{\mu}\tilde{h}_{2}(x)=V_0+\mu h_{2}(x)$, for all $x\in \mathbb{R}^3$.

 Throughout this paper, instead of (A1) we make the following assumptions:
\begin{itemize}
\item[(A2)] $\tilde{V}\in C(\mathbb{R}^3,\mathbb{R})$ and 
$\inf_{x\in \mathbb{R}^3}\tilde{V}(x)\geq a>0$, where $a>0$ is a constant, 
and there exists a constant $d_0>0$ such that
$$
\lim_{|y|\to +\infty}\operatorname{meas}\{x\in \mathbb{R}^3:|x-y|
\leq d_0,\;\tilde{V}(x)\leq M\}=0, \quad \forall M>0.
$$
\end{itemize}
Then it is easy to verify the following lemma.

\begin{lemma} \label{lem1.1} 
System \eqref{e1.1} is equivalent to the  problem
\begin{equation} \label{e1.2}
 \begin{gathered}
 -\Delta u+\tilde{V}(x)u+k(x)\phi(x)u=h_1(x)|u|^{4}u
 +\tilde{\mu} \tilde{h}_{2}(x)u+h_3(x), \quad\text{in } \mathbb{R}^3,\\
 -\Delta \phi(x)=k(x)u^2 ,\quad\text{in }\mathbb{ R}^3.
 \end{gathered}
 \end{equation}
\end{lemma}

We also assume that
\begin{itemize}
\item[(A3)] $k\in L^{\infty}(\mathbb{R}^3,\mathbb{R})$, and $k(x)\geq0$ 
for any $x\in \mathbb{R}^3$.

\item[(A4)] $h_1, h_3\in L^2\cap C_0^{\infty}(\mathbb{R}^3,\mathbb{R})$
and $\tilde{h}_{2}\in L^{6}\cap C_0^{\infty}(\mathbb{R}^3,\mathbb{R})$.

(A5) $0<\tilde{\mu}<\mu_0$, where $\mu_0$ is defined by
\begin{equation}
\mu_0:=\inf_{u\in H^{1}(\mathbb{R}^3)\setminus\{0\}}
\Big\{\int_{\mathbb{R}^3}(|\nabla u|^2+\tilde{V}(x)u^2)\mathrm{d}x:
\int_{\mathbb{R}^3}\tilde{h}_{2}(x)|u|^2\mathrm{d}x=1\Big\}.  \label{e1.3}
\end{equation}
\end{itemize}

\begin{proposition}[{\cite[Lemma 2.5]{h1}}] \label{prop1.2} 
 Assume that {\rm (A3)}  and {\rm (A4)}  hold.
 Then the  infimum $\mu_0$  is achieved.
\end{proposition}

Now we state our main results.

\begin{theorem} \label{thm1.3}
 Suppose that {\rm (A2)--(A5)} hold. Then there exists
 $m_0>0$  such that  \eqref{e1.2} admits at least two nontrivial weak
 solutions when $\|h_3\|_{2}\leq m_0$. Actually, one solution is positive 
and one is negative.
\end{theorem}

 \begin{remark} \label{rmk1.4}\rm
 It is not difficult to find the functions $V(x)$ satisfying the above conditions.
 For example, let $V(x)$ be a zig-zag function with respect to $|x|$ defined as
\[
V(x)= \begin{cases}
2n|x|-2n(n-1)+a_0, & n-1\leq|x|\leq\frac{(2n-1)}{2},  \\
-2n|x|+2n^2+a_0, & \frac{(2n-1)}{2}\leq|x|\leq n,
\end{cases}
\]
where $n\in\mathbb{ N}$ and $a_0\in \mathbb{R}$. Set 
$V_0:=\sup_{x\in\mathbb{ R}}|V(x)|$, it is not difficult to verify that 
$V(x)$ satisfies the conditions (A1) and (A2).
\end{remark}

\begin{remark} \label{rmk1.5} \rm
The nonlinear growth of $|u|^{4}u$ reaches the Sobolev critical exponent 
since the critical exponent $2^{\ast}=6$ in three spatial dimensions, 
which is why we call it critical nonlinearity in the title.
\end{remark}

The remainder of this paper is organized as follows. 
In Section 2, some preliminary results are presented. 
In Section 3, we give the proofs of our main results.

Hereafter, we use the following notation.\\
$\bullet$ $H^{1}(\mathbb{R}^3)$ denotes the usual sobolev space endowed with 
the standard scalar product and norm 
$$
(u,v)=\int_{\mathbb{R}^3}(\nabla u\nabla v+uv)\mathrm{d}x,\quad
\|u\|^2=\int_{\mathbb{R}^3}(|\nabla u|^2+u^2)\mathrm{d}x.
$$
$\bullet$ $D^{1,2}(\mathbb{R}^3)$ is the completion of 
$C^{\infty}_0(\mathbb{R}^3)$ with respect to the norm 
$\|u\|_{D^{1,2}}:=(\int_{\mathbb{R}^3}|\nabla u|^2\mathrm{d}x)^{1/2}$.
\\
$\bullet$ Denote the space
$E=\{u\in H^{1}(\mathbb{R}^3):\int_{\mathbb{R}^3}
(|\nabla u|^2+\tilde{V}(x)|u|^2)\mathrm{d}x<\infty\}$, with the norm
$$
\|u\|_{E}^2=\int_{\mathbb{R}^3}(|\nabla u|^2+\tilde{V}(x)|u|^2)
\mathrm{d}x.
$$
$\bullet$ $E^{\ast}$ denotes the dual space of $E$.\\
$\bullet$ For any $\rho>0$ and for any $z\in \mathbb{R}^3$, 
$B_{\rho(z)}$ 
denotes the ball of radius $\rho$ centered at $z$. 
$|B_{\rho(z)}|$ denotes its lebesgue measure of the ball.

\section{Variational setting and preliminaries}

In this section, we recall some basic notation and preliminaries.
From [3], we know that under the assumption (A2), the embedding
$E\hookrightarrow L^{s}(\mathbb{R}^3)$ is compact for $s\in[2,6)$.

It is easy to show that system \eqref{e1.2}
can be reduced to a single equation with a nonlocal term.
For $u\in E$, we define a linear functional
$L_{u}$ in $D^{1,2}(\mathbb{R}^3)$ as follows:
$$
L_{u}:v\to \int_{\mathbb{R}^3}k(x)u^2v\mathrm{d}x.
$$
One can check that the functional $L_{u}$ is continuous in
$D^{1,2}(\mathbb{R}^3)$.
Indeed, by using the H\"older's inequality and Sobolev inequality,
we obtain 
\begin{equation}
\big|\int_{\mathbb{R}^3}k(x)u^2v\mathrm{d}x\big|
\leq \|k\|_{\infty}\|u^2\|_{\frac{6}{5}}\|v\|_{6}
= \|k\|_{\infty} \|u\|_{\frac{12}{5}}^2\|v\|_{6}
\leq C\|u\|_{E}^2\|v\|_{D^{1,2}}. \label{e2.1}
\end{equation}
Given $u\in E$, by  the Lax-Milgram Theorem,
there exists a unique solution $\phi_{u}\in D^{1,2}(\mathbb{R}^3)$
of the equation
$$
-\Delta \phi(x)=k(x)u^2.
$$
Moreover, $\phi_{u}$ has the  integral expression
\[
\phi_{u}(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}k(y)\frac{u^2(y)}{|x-y|}
\mathrm{d}y\geq0,
\]
combining this with \eqref{e2.1}, we obtain
$$
\|\phi\|_{D^{1,2}}^2<C\|u\|_{E}^2\|\phi_{u}\|_{D^{1,2}};
$$
that is, $\|\phi_{u}\|_{D^{1,2}}\leq c\|u\|_{E}^2$.
And then we have
\begin{equation}
\begin{aligned}
 \frac{1}{4\pi}\int_{\mathbb{R}^3}
 \int_{\mathbb{R}^3}
 k(y)\frac{u^2(x)u^2(y)}{|x-y|}\mathrm{d}x\mathrm{d}y
 &=\int_{\mathbb{R}^3}u^2\phi_{u}(x)\mathrm{d}x \\
 &\leq C\|u\|_{E}^{4}.\label{e2.2}
 \end{aligned}
\end{equation}
Now we define a functional $I$ on $E$ by
 \begin{align*}
 I(u)=&\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+\tilde{V}(x)u^2)\mathrm{d}x
 +  \frac{1}{4}\int_{\mathbb{R}^3}k(x)\phi_{u}u^2\mathrm{d}x \\
 &-  \frac{1}{6}\int_{\mathbb{R}^3}h_1(x)u^{6}\mathrm{d}x
 -  \frac{1}{2}\int_{\mathbb{R}^3}\tilde{\mu}\tilde{h}_{2}(x)u^2\mathrm{d}x
 -  \int_{\mathbb{R}^3}h_3(x)u\mathrm{d}x.
 \end{align*}
 It is easy to verify that the functional  $I$ is of class
 $C^{1}(E,\mathbb{R})$.  Moreover,
 \begin{align*}
 \langle I'(u),v\rangle
&=\int_{\mathbb{R}^3}(|\nabla u|\nabla v+\tilde{V}(x)uv)\mathrm{d}x
 +  \int_{\mathbb{R}^3}k(x)\phi_{u}uv\mathrm{d}x  \\
&\quad  - \int_{\mathbb{R}^3}h_1(x)u^{5}v\mathrm{d}x
- \int_{\mathbb{R}^3}\tilde{\mu}\tilde{h}_{2}(x)uv\mathrm{d}x
-  \int_{\mathbb{R}^3}h_3(x)v\mathrm{d}x.
 \end{align*}
Hence, if $u\in E$ is a critical point of $I$, then the pair
$(u,\phi_{u})$ is a solution of \eqref{e1.2}.

\begin{theorem}[{\cite[Mountain Pass Theorem]{w2}}] \label{thm2.1}
Let $X$ be a real Banach space, suppose that $I\in C^{1}(X,\mathbb{R})$
satisfies the (PS) condition with $I(0)=0$.
In addition, suppose that
\begin{itemize}
\item[(i)] there are $\rho, \alpha>0$ such that
$I(u)\geq\alpha$  when $\|u\|_{X}=\rho$;

\item[(ii)] there is $e\in X$, $\|e\|_{X}>\rho$, such that $I(e)<0$.
\end{itemize}
Define
$$
\Gamma=\{\gamma\in C^{1}([0,1],X)|\gamma(0)=0, \gamma(1)=e\}.
$$
Then
$c:=\inf_{\gamma\in\Gamma}\max_{t\in[0,1]}I(\gamma(t))$
is a critical value of $I$.
\end{theorem}

\section{Proof of Theorem \ref{thm1.3}}\label{section3}

\begin{lemma} \label{lem3.1}
Assume that {\rm (A2)--(A5)} hold.
Then $I$ satisfies the (PS) condition.
\end{lemma}

\begin{proof}
We first prove that $\{u_{n}\}$ is bounded in $E$. Then
\begin{equation} \label{e3.1}
\begin{aligned}
C+1+\|u_{n}\|_{E} 
&\geq I(u_{n})-\frac{1}{6}\langle I'(u_{n}),u_{n}\rangle \\
&=\frac{1}{3}\|u_{n}\|^2_{E}
 +\frac{1}{12}\int_{\mathbb{R}^3}k(x)\Phi u^2\mathrm{d}x \\
&\quad -\frac{1}{3}\tilde{\mu}\int_{\mathbb{R}^3}\tilde{h}_{2}(x)u^2\mathrm{d}x-
 (1-\frac{1}{6})\int_{\mathbb{R}^3}h_3(x)u_{n}\mathrm{d}x \\
&\geq\frac{1}{3}\|u_{n}\|^2_{E}
 +\frac{1}{12}\int_{\mathbb{R}^3}k(x)\Phi u^2_{n}\mathrm{d}x
 -\frac{\tilde{\mu}}{3\mu_0}\|u_{n}\|^2_{E} \\
&\quad - \frac{5}{6}\Big(\int_{\mathbb{R}^3}h^2_3(x)\mathrm{d}x\Big)^{1/2}
\Big(\int_{\mathbb{R}^3}u^2_{n}\mathrm{d}x\Big)^{1/2} \\
&\geq\frac{1}{3}(1-\frac{\tilde{\mu}}{\mu_0})\|u_{n}\|^2_{E}-C\|u_{n}\|_{E},
\end{aligned}
\end{equation}
which implies that $\{u_{n}\}$ is bounded.

Next we show that $\{u_{n}\}$ possesses a strong convergent subsequence in $E$.
In fact, in view of the boundedness of $\{u_{n}\}$,
without loss of generality, we assume that there exists
$u_0\in E$ such that
$u_{n}\rightharpoonup u_0$ as $n\to \infty$.
Since $E\hookrightarrow L^{s}(\mathbb{R}^3)$ is compact for $s\in[2,6)$, 
$u_{n}\to  u_0$ in $L^{s}(\mathbb{R}^3)$ for any $s\in[2,6)$.
We obtain
\begin{equation} \label{e3.2}
\begin{aligned}
\|u_{n}-u\|^2_{E}
&= 2 \langle I'(u_{n})-I'(u),u_{n}-u\rangle
-\int_{\mathbb{R}^3}(\phi_{u_{n}}u_{n}-\phi_{u}u)(u_{n}-u)\mathrm{d}x  \\
&\quad +\int_{\mathbb{R}^3}h_1(|u_{n}|^{4}u-|u|^{4}u)(u_{n}-u)\mathrm{d}x
+\tilde{\mu}\int_{\mathbb{R}^3}\tilde{h}_{2}(x)(u_{n}-u)^2\mathrm{d}x  \\
&\quad +\int_{\mathbb{R}^3}h_3(x)(u_{n}-u)\mathrm{d}x.
\end{aligned}
\end{equation}
It is clear that
\[
\langle I'(u_{n})-I'(u),u_{n}-u\rangle\to 0.
\]
By using the H\"older's inequality and Sobolev inequality,
we obtain
\begin{equation} \label{e3.4}
\begin{aligned}
\big|\int_{\mathbb{R}^3}\phi_{u_{n}}u_{n}(u_{n}-u)\mathrm{d}x\big|
 &\leq\|\phi_{u_{n}}u_{n}\|_{2}\|u_{n}-u\|_{2}\\
 &\leq\|\phi_{u_{n}}\|_{6}\|u_{n}\|_3\|u_{n}-u\|_{2} \\
 &\leq C\|\phi_{u_{n}}\|_{D^{1,2}}\|u_{n}\|_3\|u_{n}-u\|_3\\
 &\leq C\|u_{n}\|_{\frac{12}{5}}\|u_{n}\|_3\|u_{n}-u\|_{2}\to 0.
\end{aligned}
\end{equation}
Similarly,
we obtain that
$\int_{\mathbb{R}^3}\phi_{u}u(u_{n}-u)\mathrm{d}x\to 0$ as $n\to \infty$.

From the Br\'{e}zis-Lieb Lemma \cite{w2}, we have
\begin{equation} \label{e3.5}
 \begin{aligned}
&\big|\int_{\mathbb{R}^3}h_1(x)(|u_{n}|^{5}-|u|^{5})(u_{n}-u)\mathrm{d}x\big|\\
 &\leq\|h_1\|_{\infty}
\big|\int_{\mathbb{R}^3}(|u_{n}|^{5}-|u^{5}|)(u_{n}-u)\mathrm{d}x\big|\\
 &\leq\|h_1\|_{\infty}\int_{\mathbb{R}^3}|u_{n}-u|^{6}\mathrm{d}x+o(1)\\
 &\leq C\|h_1\|_{\infty}\|u_{n}-u\|^{6}_{E}+o(1)\to 0.
 \end{aligned}
\end{equation}
By using the H\"older's inequality,
we obtain
 \begin{gather}
\int_{\mathbb{R}^3}h_3(x)(u_{n}-u)\mathrm{d}x\leq\|h_3\|_{2}\|u_{n}-u\|_{2}\to 0,\\
\int_{\mathbb{R}^3}\tilde{h}_{2}(x)(u_{n}-u)^2\mathrm{d}x
\leq\|\tilde{h}_{2}\|_{6}\|u_{n}-u\|_{12/5}^2\to 0.
 \end{gather}
We obtain
$\|u_{n}-u\|^2_{E}\to 0$,
which completes the proof.
\end{proof}

\begin{lemma} \label{lem3.2}
Assume that {\rm (A2)--(A5)} are satisfied.
Then there exists $m_0>0$ such that \eqref{e1.2}
has a positive solution when $\|h_3\|_{2}\leq m_0$.
\end{lemma}

\begin{proof} 
It follows from Lemma \ref{lem1.1} and the Sobolev inequality that
\begin{align*}
I(u)
&=\frac{1}{2}\int_{\mathbb{R}^3}|\nabla u|^2\mathrm{d}x
+\frac{1}{2}\int_{\mathbb{R}^3}\tilde{V}(x)u^2\mathrm{d}x
+\frac{1}{4}\int_{\mathbb{R}^3}k(x)\phi_{u}u^2\mathrm{d}x
-\frac{1}{6}\int_{\mathbb{R}^3}h_1(x)u^{6}\mathrm{d}x \\
&\quad - \frac{1}{2}\int_{\mathbb{R}^3}\tilde{\mu}\tilde{h}_{2}(x)u^2\mathrm{d}x
- \int_{\mathbb{R}^3}h_3(x)u\mathrm{d}x \\
&=\frac{1}{2}\|u\|_{E}^2
+\frac{1}{4}\int_{\mathbb{R}^3}k(x)\phi_{u}u^2\mathrm{d}x
-\frac{1}{6}\int_{\mathbb{R}^3}h_1(x)u^{6}\mathrm{d}x \\
&\quad -\frac{1}{2}\tilde{\mu}\int_{\mathbb{R}^3}\tilde{h}_{2}(x)u^2\mathrm{d}x
-\int_{\mathbb{R}^3}h_3(x)u\mathrm{d}x\\
&\geq \frac{1}{2}\|u\|_{E}^2
-\frac{C}{6}\|h_1\|_{\infty}\|u\|_{E}^{6} 
 -\frac{1}{2}\tilde{\mu}\int_{\mathbb{R}^3}|\tilde{h}_{2}(x)|u^2\mathrm{d}x
-\|h_3\|_{2}\|u\|_{2}\\
&\geq \frac{1}{2}(1-\frac{\tilde{\mu}}{\mu_0})\|u\|_{E}^2
-\frac{C}{6}\|h_1\|_{\infty}\|u\|_{E}^{6}
-C_1\|h_3\|_{2}\|u\|_{E}\\
&=\|u\|_{E}\Big(\frac{1}{2}(1-\frac{\tilde{\mu}}{\mu_0})\|u\|_{E}
-\frac{C}{6}\|u\|_{E}^{5}
-C_1\|h_3\|_{2}\Big).
\end{align*}
Taking $\|u\|_{E}=t$, and letting
$$
g(t)=\frac{1}{2}(1-\frac{\tilde{\mu}}{\mu_0})t-\frac{C}{6}t^{5},
$$
we have $g'(t_0)=0$ when
$$
t_0=\Big(\frac{3(1-\frac{\tilde{\mu}}{\mu_0})}{5C}\Big)^{1/4}.
$$
Hence,
there exists
$$
m_0=\|h_3\|_{2}<\frac{2}{5C_1}(1-\frac{\tilde{\mu}}{\mu_0})
\Big(\frac{3(1-\frac{\tilde{\mu}}{\mu_0})}{5C}\Big)^{1/4},
$$
such that
$$
\frac{2}{5}(1-\frac{\tilde{\mu}}{\mu_0})
\Big(\frac{3(1-\frac{\tilde{\mu}}{\mu_0})}{5C}\Big)^{1/4}
-C_1\|h_3\|_{2}>0.
$$
Let $\rho=\|u\|_{E}$ small enough, there exists $\alpha>0$,
such that $I(u)\geq\alpha$. Thus,
(i) in Theorem \ref{thm2.1} is true.

Choose $\psi_0\in C_0^{\infty}(\mathbb{R}^3), \psi_0\geq0$ and
$\psi_0\not\equiv0$ in $\mathbb{R}^3$. Then
\begin{equation}
\begin{aligned}
I(t\psi_0)
&=\frac{1}{2}t^2\|\psi_0\|^2
+\frac{1}{4}t^{4}\int_{\mathbb{R}^3}k(x)\phi_{\psi_0}\psi_0^2\mathrm{d}x
-\frac{1}{6}t^{6}\int_{\mathbb{R}^3}h_1(x)\psi_0^{6}\mathrm{d}x \\
&\quad -\frac{1}{2}t^2\tilde{\mu}\int_{\mathbb{R}^3}h_{2}(x)\psi_0^2\mathrm{d}x
-t\int_{\mathbb{R}^3}h_3(x)\psi_0\mathrm{d}x,
\end{aligned}
\end{equation}
so $I(t\psi_0)\to -\infty$ as $t\to +\infty$.
Therefore, there exists $t_0$ large enough, such that $I(t_0\psi_0)<0$.
Taking $e=t_0\psi_0$ such that $\|t_0\psi_0\|>\rho$
and $I(e)<0$, and (ii) in Theorem \ref{thm2.1} is true.
It is obvious that $I(0)=0$, by Theorem \ref{thm2.1},
problem \eqref{e1.2} has a positive solution.
\end{proof}

\begin{lemma} \label{lem3.3}
Assume that {\rm (A2)--(A4)} hold.
Then \eqref{e1.2} has a negative solution.
\end{lemma}

\begin{proof}
Since $h_3\in L^2(\mathbb{R}^3)\setminus\{0\}$ and $h^{+}_3\not\equiv0$,
we can choose a function $\phi_1\in E$ such that
$$
\int_{R^3}h_3(x)\phi_1\mathrm{d}x>0.
$$
For $t>0$ small enough, we have
\begin{align*}
I(t\phi_1)
&=\frac{t^2}{2}\|\phi_1\|^2_{E}
+\frac{t^2}{4}k(x)\int_{\mathbb{R}^3}\phi_{u}\phi^2_1\mathrm{d}x
-\frac{t^{6}}{6}\int_{\mathbb{R}^3}h_1(x)\phi_1^{6}\mathrm{d}x \\
&\quad -\frac{\tilde{\mu}t^2}{2}\int_{\mathbb{R}^3}\tilde{h}_{2}(x)
\phi_1\mathrm{d}x
- t\int_{\mathbb{R}^3}h_3\phi_1\mathrm{d}x  \\
&\leq\frac{t^2}{2}\|\phi_1\|^2_{E}
+\frac{t^{4}}{4}c\|\phi_1\|^{4}_{E}
-\frac{t^{6}}{6}\int_{\mathbb{R}^3}h_3(x)\phi^{6}_1\mathrm{d}x
-\frac{\tilde{\mu}t^2}{2}\int_{\mathbb{R}^3}\tilde{h}_{2}(x)\phi_1\mathrm{d}x \\
&\quad -t\int_{\mathbb{R}^3}h_3(x)\phi_1\mathrm{d}x 
<0.
\end{align*}
Hence $\theta_0:=\inf\{I(u):u\in\bar{B_{\rho}}\}<0$.
By the Ekeland's variational principle,
there exists a minimizing sequence $\{u_{n}\}\subset\bar{B_{\rho}}$,
such that $I(u_{n})\to 0$ and
$I'(u_{n})\to 0$ as $n\to \infty$.
Because the functional $I$ satisfies the (PS) condition,
there exists $u_0\in E$ such that
$I'(u_0)=0$ and $I(u_0)=c_1<0$.
The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
From Lemmas \ref{lem3.2} and \ref{lem3.3}, we obtain the existence of at least two nontrivial
 weak solutions for the problem \eqref{e1.2}.
Actually, one  solution is positive and the other negative.
\end{proof}

\subsection*{Acknowledgments}
This work is partially supported by the Natural Science Foundation of China 11671403.


 \begin{thebibliography}{00}

\bibitem{a1} A. Ambrosetti, D. Ruiz;
 \newblock{Multiple bound states for the schr\"odinger-Poisson problem,}
\newblock {\em Commun. Contemp. Math.} \textbf{10} (2008), 391-404.


\bibitem{a2} A. Azzollini, A. Pomponio;
\newblock{Ground state solutions for the nonlinear Schr\"odinger-Maxwell equations,}
\newblock{\em J. Math. Anal. Appl.} \textbf{345} (2008), 90-108.

\bibitem{b1} T. Bartsch, Z. Wang, M. Willem;
\newblock{The Dirichlet problem for superlinear elliptic equations, 
in: M.chipot, P. Quittner (Eds.),}
\newblock{\em Handbook of Differential Equations-stationary Partial Differential 
Equations. Vol. 2. Elsevier}  (2005) .

\bibitem{c1}  I. Catto, P. Lions;
 \newblock{Binding of atoms and stability of molecules in Hartree and Thomas-Fermi 
type theories. Part 1: Anessary and sufficient condition for the stability of 
general molecular system,}
 \newblock{\em Comm. Partial Differential Equations.} \textbf{17 }(1992), 1051-1110.

\bibitem{c2} G. Cerami, G. Vaira;
 \newblock{Positive solutions for some non-autonomous Schr\"odinger-Poisson
systems,} \newblock {\em J. Differential Equations.} \textbf{248} (2010), 521-543.

\bibitem{h1} L. Huang, E. M. Rocha;
\newblock{A positive solution of a Schr\"odinger-Poisson system with critical
exponent,} \newblock{\em Commun. Math. Appl.} \textbf{15} (2013), 29-43.

\bibitem{j1} Y. Jiang, Z. Wang, H. Zhou;
\newblock{Multiple solutions for a nonhomogeneous Schr\"odinger-Maxwell system
in $\mathbb{R}^3$,}
\newblock {\em Nonlinear. Anal.} \textbf{83} (2013), 50-57.

\bibitem{j2}  Y. Jiang, H. Zhou;
 \newblock{Schr\"odinger-Poisson system with steep potential well, }
\newblock {\em J. Differential Equations.} \textbf{251} (2011), 582-608.


\bibitem{l1} Q. Li, H. Su, Z. Wei;
\newblock{Existence of infinitely many large solutions for the nonlinear Schr\"odinger-Maxwell equations,}
\newblock{\em Nonlinear. Anal.} \textbf{72} (2010) 4264-4270.


\bibitem{l2} Y.Q. Li, Z.Q. Wang, J. Zeng;
 \newblock{Ground states of nonlinear Schr\"odinger equations with potentials,}
 \newblock{ \em Ann. Inst. HenriPoincar\'{e}, Anal. Non Lineair\'{e}.} 
\textbf{23} (2006), 829-837.

\bibitem{l3} Z. P. Liang, J. Xu, X. L. Zhu;
\newblock{Revisit to sign-changing solutions for nonlinear Schr\"odinger-Poisson
 system $\mathbb{R}^3$,}
\newblock {\em J. Math. Anal. Appl.} \textbf{435} (2016), 783-799.

\bibitem{l4} E. H. Lieb;
\newblock{Thomas-Fermi and related theories and molecules,}
\newblock{\em Rev. Modern. Phs.} \textbf{53} (1981), 603-641.

\bibitem{l5}  P. Lions;
 \newblock{Solutions of Hartree-Fock equations for coulomb system, }
\newblock {\em Comm. Math. Phys.} \textbf{109} (1984), 33--97.


\bibitem{l6} Z. Liu, S. Guo;
\newblock{On ground state solutions for the Schr\"odinger-Poisson equations
with critical growth,}  \newblock{\em J. Math. Anal. Appl.}
 \textbf{412} (2014), 435-448.


\bibitem{r1}  P. H. Rabinowitz;
\newblock{On a class of nonlinear Schr\"odinger equations,}
 \newblock{\em Z. Angew. Math. Phys.} \textbf{43} (1992), 270-291.

\bibitem{s1} H. X. Shi, H. Chen;
\newblock{Generalized quasilinear asymptotically periodic Schr\"odinger
equations with critical growth,}
\newblock{\em Commun. Math. Appl.} \textbf{71} (2016), 849-858.

\bibitem{s2} H. X. Shi, H. Chen;
\newblock{Ground state solutions for asymptotically
periodic coupled Kirchhoff-type systems with critical growth,}
\newblock{\em Math. Meth. Appl. Sci.} \textbf{39} (2016), 2193-2201.

\bibitem{s3} J. J. Sun, S. W. Ma;
\newblock{Ground state solutions for some Schr\"odinger-Poisson systems
with periodic potentials,}
\newblock{\em J. Differential Equations.} \textbf{68} (2015), 201-216.

\bibitem{s4} J. Sun, H. Chen, J. J. Nieto;
\newblock{On ground state solutions for some non-autonomous Schr\"odinger-Poisson
 system,} \newblock{\em J. Differential Equations.} \textbf{252} (2012), 3365-3380.

\bibitem{s5} J. T. Sun, T. F. Wu, Z. S. Feng;
\newblock{Multipicity of positive solutions for a nonlinear Schr\"odinger-Poisson
 system,} \newblock{\em J. Differential Equations.} \textbf{260} (2016), 586-627.


\bibitem{s6}  A. Szulkin, T. Weth;
 \newblock{Ground state solutions for some indefinite variational problems,}
 \newblock{\em J. Funct. Anal.}\textbf{ 257} (2009), 3802-3822.

\bibitem{t1} X. H. Tang;
\newblock{Infinitely many solutins for semilinear Schr\"odinger equation
 with sign-changing potential and nonlinearity,} 
\newblock{\em J. Math. Anal. Appl.} \textbf{401} (2013), 407-415.

\bibitem{t2} X. H. Tang;
\newblock{New conditions on nonlinearity for a periodic Schr\"odinger
equation having zero as spectrum,}
\newblock{\em J. Math. Anal. Appl.} \textbf{413} (2014), 392-410.

\bibitem{t3} X. H. Tang;
\newblock{New super-quadratic conditions on ground state solutions for 
superlinear Schr\"odinger equation,}
\newblock{\em Adv. Nonlinear Studies.} \textbf{14} (2014), 349-361.

\bibitem{w1} Z. Q. Wang;
\newblock{Nonlinear boundary value problems with concave nonlinearities near 
the origin,} \newblock{\em Nonlinear Differential Equations Appl.} 
\textbf{8} (2001), 15-33.

\bibitem{w2} M. Willem;
\newblock{Minimax Theorems,}
 \newblock{\em Birkh\"ouser, Berlin,} 1996.

\bibitem{z1} Q. Zhang, B. Xu;
\newblock{Multiplicity of solutions for a class of semilinear Schr\"odinger
equations with sign-changing potenial,}
\newblock{\em J. Math. Anal. Appl.} \textbf{377} (2011), 834-840.

\end{thebibliography}

\end{document}