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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011(2011), No. 01, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/01\hfil Schr\"odinger-Poisson equations]
{Schr\"odinger-Poisson equations with supercritical growth}

\author[C. O. Alves, S. H. M. Soares, M. A. S. Souto\hfil EJDE-2011/01\hfilneg]
{Claudianor O. Alves, S\'ergio H. M. Soares, Marco A. S. Souto}
  % in alphabetical order

\address{Claudianor O. Alves \newline
 Unidade Acad\^emica de Matem\'atica e Estat\'istica,
 Universidade Federal de Campina Grande,
 58109-970, Campina Grande - PB, Brazil}
\email{coalves@dme.ufcg.edu.br}

\address{S\'ergio H. M. Soares \newline
 Departamento de Matem\'atica \\
 Instituto de Ci\^encias Matem\'{a}ticas e de  Computa\c{c}\~ao\\
 Universidade de S\~ao Paulo \\
 13560-970, S\~ao Carlos - SP, Brazil}
\email{monari@icmc.usp.br}

\address{Marco A. S. Souto \newline
 Unidade Acad\^emica de Matem\'atica e Estat\'istica,
 Universidade Federal de Campina Grande,
 58109-970, Campina Grande - PB, Brazil}
\email{marco@dme.ufcg.edu.br, Phone +5583-3310-1501, Fax +5583-3310-1112}

\thanks{Submitted June 1, 2010. Published January 4, 2011.}
\thanks{C.O. Alves is supported by grants 620150/2008-4 and
303080/2009-4 from CNPq/Brazil.\hfill\break\indent
S.H.M. Soares was supported by grant 313237/2009-3 from  CNPq/Brazil.
\hfill\break\indent
 M.A.S. Souto is  supported by grants 2009/14218-2 from 
 FAPESP, PROCAD 024/2007  \hfill\break\indent
 from CAPES/Brazil
and 302650/2008-3 from CNPq/Brazil.} 
\subjclass[2000]{35J20, 35J65}
\keywords{Supercritical problems; Schr\"odinger-Poisson equation;
\hfill\break\indent  variational method}

\begin{abstract}
 In this article, we study a class of Schr\"odinger-Poisson equations
 in $\mathbb{R}^3$ with supercritical growth.
 We prove the existence of positive solutions, using
 variational methods combined with perturbation arguments.
 The solutions to subcritical Schr\"odinger-Poisson
 equations are estimated using the $L^\infty$ norm.
\end{abstract}

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

\section{Introduction}

The system
\begin{equation} \label{eS}
\begin{gathered}
- \Delta u + V(x)u + \phi u
 = |u|^{p-2}u \quad\text{in }\mathbb{R}^3,\\
-\Delta \phi=u^2 \quad\text{in }\mathbb{R}^3
\end{gathered}
\end{equation}
has  great importance for describing the interaction of a charged
particle with an electromagnetic field; see for example
\cite{AM,ap,bf,G,Ruiz,Ruiz2}.
Recent studies have focused attention on existence,
nonexistence and symmetry of solutions to \eqref{eS}.
However, most of the references presented here
are devoted to pure power type nonlinearities $|u|^{p-2}u$,
where  $p$ is  a subcritical or critical exponent.
For example, Ruiz \cite{Ruiz} studies the subcritical case
and shows that  if $p\leq 3$, the problem \eqref{eS} does not
admit any   nontrivial solution, and if $3<p<6$, there exists
a nontrivial   radial solution of \eqref{eS}.
A multiplicity result for the subcritical case has been
established by  Gaetano \cite{G}.
The critical case has been treated by Azzollini and
Pomponio \cite{ap} and Zhao and Zhao \cite{zz}.

In this article, we consider the stationary Schr\"odinger-Poisson
system
\begin{equation} \label{eP}
\begin{gathered}
- \Delta u +V(x)u+\phi u= f(u),  \quad\text{in } \mathbb{R}^3, \\
 -\Delta \phi=u^2, \quad\text{in } \mathbb{R}^3,
\end{gathered}
\end{equation}
where $V:\mathbb{R}^3 \to \mathbb{R}$ is a bounded locally
H\"older continuous function satisfying:
\begin{itemize}
\item[(V0)]  There exists $\alpha >0$ such that
$V(x) \geq \alpha >0$, $\forall x \in \mathbb{R}^3$.

\item[(V1)]  $V(x)=V(x+y)$,   for all $x \in \mathbb{R}^3$,
$y\in \mathbb{Z}^3$.
\end{itemize}

The function  $f\in C(\mathbb{R},\mathbb{R})$ can be written as
$$
f(s)=f_o(s)+\lambda g(s),
$$
where $\lambda$ is a
positive real parameter, $f_o$ and $g$ are locally H\"older
continuous functions satisfying:
\begin{itemize}
\item[(F1)]
$f_o(0)=g(0)=0$ and $g(s)\geq 0$ for all $s$;

\item[(F2)]
$\lim_{|s|\to 0 ^+} f_o(s)/s = 0$ and
$\lim_{|s|\to 0 ^+} g(s)/s = 0$;

\item[(F3)] There exists $q \in (4,2^*)$, $2^*=6$, such that
$$
|f_o(s)|\leq |s|^{q-1},\quad \forall s \in \mathbb{R};
$$

\item[(F4)]
$\lim_{|s|\to +\infty} F_o(s)/s^4=+\infty$, where
$F_o(s)=\int_0^s f_o(t)dt$;

\item[(F5)] For $\alpha>0$ given by $(V_0)$, there exists
$\sigma \in (0,\alpha)$ such that
$$
sf_o(s)-4F_o(s)\geq -\sigma s^2\quad  \text{and}\quad
 sg(s)-4G(s)\geq 0
$$
for all $s \neq 0$, where $G(s)=\int_0^s g(t)dt$;

\item[(F6)]
There exists a sequence of positive real numbers, $(M_n)$,
converging  to  $+\infty$ such that
$$
\frac{g(s)}{s^{q-1}} \leq \frac{g(M_n)}{M_n^{q-1}},\quad
\text{for all } s \in [0,M_n],\;  n\in \mathbb{N}.
$$
\end{itemize}

Since $u\equiv0$ is a solution of \eqref{eP}, the aim of the
present article is to prove the existence of nontrivial solutions
for \eqref{eP}. However, it should be point out that we can not
apply variational methods directly  because  the Euler-Lagrange
functional on $H^1(\mathbb{R}^3)$  associated with \eqref{eP} is
not well defined in general. Our technique combines perturbation
arguments, estimate for solutions to a subcritical
Schr\"odinger-Poisson equation in terms of the $L^\infty$ norm and
the mountain pass theorem.  Our main result is as follows.

\begin{theorem} \label{thm1}
Suppose that $V$ satisfies {\rm (V0)--(V1)}, and  $f$ satisfies
{\rm (F1)--(F6)}. Then there is a $\lambda_o>0$ such that
\eqref{eP} possesses a positive  solution for all $\lambda \leq
\lambda_o$.
\end{theorem}

To prove the above theorem, we argue as in Alves and Souto
\cite{as}. We first provide an estimate involving the
$L^{\infty}$-norm of a solution related to a subcritical problem.
To do so  we  modify the nonlinearity obtaining a family of
functionals of class $C^1$. Employing conditions (F1)--(F4), we
show that these functionals satisfy uniformly  the geometric
hypotheses of the mountain pass theorem. Using this fact and the
estimate, we  verify the existence of a sequence in
$H^1(\mathbb{R}^3)$ converging weakly to a solution of \eqref{eP}.
It is important to stress that our proof does not require a growth
assumption on $g$; consequently, on $f$. We observe that the
condition (F6) holds if
$$
\lim_{|s|\to +\infty}\frac{g(s)}{s^{q-1}} = +\infty.
$$
In particular, $f$ may be $f(s)= s^{q-1} + s^{p-1}$, for all
$p>6>q$, or $f(s)$ may behave like $e^{s}$ at infinity.

In addition, since the term $\int_{\mathbb{R}^3}\phi_u u^2\,dx$ is
homogeneous of degree 4,  the corresponding Ambrosetti-Rabinowitz
condition on $f$ is the following:
\begin{itemize}
\item[(AR)] There exists $\theta >4$ such that
\[
0 < \theta F(s) \leq sf(s),\quad \forall s\in \mathbb{R}.
\]
\end{itemize}
This condition is important not only to ensure that the functional
$I$ (see \eqref{defI} below) has the mountain pass geometry, but also to guarantee that the
Palais-Smale, or  Cerami, sequences associated with $I$ are bounded.
We observe that $f(s) = s^{q-1} + \lambda(1+ \cos s)s^{p-1}$, with
$4< q < 6 \leq p$, satisfies the conditions (F1)--(F6), but not
the Ambrosetti-Rabinowitz condition. Moreover, the function $f$
considered here does not belong to any class of nonlinearities in
the above-cited papers.

 Since we intend to prove the existence of positive solutions,  we
consider $f:\mathbb{R} \to \mathbb{R}$ satisfying (F1)--(F6) on
$[0,+\infty)$ and defined as zero on $(-\infty,0]$.

\section{A solution estimate}
This section we obtain an estimate involving the
$L^\infty$ norm of a solution to a subcritical problem. This result
works for any dimension $N\geq 3$.

\begin{proposition}\label{prop1}
Let $v\in H^1(\mathbb{R}^N)$ be a weak solution of the problem
\begin{equation}\label{4}
-\Delta v +b(x)v= h(x,v) , \text{in } \mathbb{R}^N,
\end{equation}
where $h: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}^N$ is a
continuous functions verifying, for some $2<q<2^*=2N/(N-2)$,
$|h(x,s)| \leq 2|s|^{q-1}$, for all $s>0$, and $b$ is a non-negative
function in $\mathbb{R}^N$. Then, for all $C>0$, there exists a
constant $k=k(q,C)>0$ such that if
$ \|v\|^2 \leq C$, then $ \|v\|_\infty \leq k$.
\end{proposition}

\begin{proof}
 For each $m\in \mathbb{N}$ and $\beta>1$,
consider
\begin{gather*}
A_m=\{x \in \mathbb{R}^N: |v|^{\beta-1} \leq m\},\quad
B_m=\mathbb{R}^N \setminus A_m,\\
v_{m}= \begin{cases} v|v|^{2(\beta-1)},& \text{in } A_m, \\
m^2v, &\text{in } B_m.
\end{cases}
\end{gather*}
Observe that $v_{m}\in H^{1}(\mathbb{R}^N )$,   $v_{m}\leq
|v|^{2\beta-1}$ and
\begin{equation}
\nabla v_{m}=(2\beta-1)|v|^{2(\beta-1)}\nabla v \quad\text{in } A_m ,
\quad
\nabla v_m = m^2\nabla v \quad \text{in } B_m.  \label{6}
\end{equation}
Using  $v_m$ as a test function in \eqref{4}, we obtain
\[
\int_{\mathbb{R}^N }(\nabla v\nabla v_{m}+b(x)vv_m)\,dx=\int_{\mathbb{R}^N }h(x,v)v_{m}\,dx.
\]
 From \eqref{6},
\begin{equation}
\int_{\mathbb{R}^N }\nabla v\nabla v_{m}\,dx=(2\beta-1)\int_{A_m
}|v|^{2(\beta-1)}|\nabla v|^{2}\,dx+m^2\int_{B_m}|\nabla v|^2\,dx.
\label{7}
\end{equation}
Let
\[
\omega_m= \begin{cases}
v|v|^{(\beta-1)}, &\text{in } A_m, \\
 mv,  &\text{in } B_m.
\end{cases}
\]
Then
\[
\omega_m^2=vv_m \leq |v|^{2\beta}, \quad
  0\leq b(x)\omega_m^2=b(x)vv_m, \quad \text{in }\mathbb{R}^N
\]
and
\begin{equation}
\nabla \omega_m=\beta|v|^{(\beta-1)}\nabla v\quad \text{ in } A_m ,
\quad  \nabla \omega_m = m\nabla v\quad  \text{in } B_m.  \label{8}
\end{equation}
Hence,
\begin{equation}
\int_{\mathbb{R}^N}|\nabla \omega_m|^2\,dx=\beta^2\int_{A_m
}|v|^{2(\beta-1)}|\nabla v|^{2}\,dx+m^2\int_{B_m}|\nabla v|^2 \,dx.
\label{8+}
\end{equation}
 From \eqref{7}-\eqref{8+}, we obtain
\begin{align*}
&\int_{\mathbb{R}^N }(|\nabla
\omega_{m}|^2+b(x)\omega_m^2)\,dx-\int_{\mathbb{R}^N }(\nabla v\nabla
v_{m}+b(x)vv_m)\,dx  \\
&=(\beta-1)^2\int_{A_m }|v|^{2(\beta-1)}|\nabla v|^{2}\,dx.
\end{align*}
 From \eqref{7} and $b(x)\geq 0$, we have
\[
(2\beta-1)\int_{A_m }|v|^{2(\beta-1)}|\nabla v|^{2}\,dx \leq
\int_{\mathbb{R}^N }(\nabla v\nabla v_{m}+b(x)vv_m)\,dx,
\]
and consequently
\[
\int_{\mathbb{R}^N }(|\nabla \omega_{m}|^2+b(x)\omega_m^2)\,dx\leq
[\frac {(\beta -1)^2}{2\beta-1}+1] \int_{\mathbb{R}^N }(\nabla
v\nabla v_{m}+b(x)vv_m)\,dx.
\]
Since   \eqref{4} holds for $v$, we have
\[
\int_{\mathbb{R}^N }(|\nabla
\omega_{m}|^2+b(x)\omega_m^2)\,dx\leq\frac{\beta^2}{2\beta-1}\int_{\mathbb{R}^N} h(x,v)v_m\,dx
\]
and
\begin{equation}
\int_{\mathbb{R}^N }(|\nabla \omega_{m}|^2+b(x)\omega_m^2)\,dx\leq
\beta^2 \int_{\mathbb{R}^N }h(x,v)v_m\,dx, \label{9}
\end{equation}
because $2\beta-1 >1$. Let E denote the  Sobolev space
\[
E=\big\{u\in H^1(\mathbb{R}^N): \int_{\mathbb{R}^N} b(x)u^2\,dx< \infty \big\}
\]
endowed with the norm
\[
\|u\|^2 = \int_{\mathbb{R}^N}(|\nabla u|^2+ b(x)u^2)\,dx.
\]
Throughout the proof, $r$ denotes $2^*=2N/(N-2)$.
Let $S$ be the best constant of the Sobolev immersion of
$H^1(\mathbb{R}^N)$ in $L^r(\mathbb{R}^N)$. Thus,
\[
\|u\|_r^2 \leq S \int_{\mathbb{R}^N}|\nabla u|^2\,dx
\]
for every $u \in H^1(\mathbb{R}^N)$.   From  \eqref{9} , since $|h(x,s)| \leq
2|s|^{q-1}$  for all $s>0$, we have
\[
\Big[ \int_{A_m}|\omega_m|^{r}\,dx \Big]^{2/r}
\leq \Big[\int_{\mathbb{R}^N}|\omega_m|^{r}\,dx \Big]^{2/r}
\leq S\beta^2 \int_{\mathbb{R}^N} h(x,v)v_m \,dx.
\]
Observing that
\[
 h(x,v)v_m=\frac{h(x,v)}{v}vv_m = \frac{h(x,v)}{v}w_m^2,
\]
we obtain
\[
\Big[ \int_{A_m}|\omega_m|^{r}\,dx \Big]^{2/r}
\leq 2S\beta^2 \int_{\mathbb{R}^N}|v|^{q-2}\omega_m^2 \,dx.
\]
For $q_1$ such that  $1/q_1+(q-2)/r=1$, it then follows
from H\"older's inequality that
\[
\Big[ \int_{A_m}|\omega_m|^{r}\,dx \Big]^{2/r} \leq
S\beta^2 \|v\|_r^{q-2} \Big[\int_{\mathbb{R}^N}|\omega_m|^{2q_1}
\,dx\Big]^{1/q_1}.
\]
Since $|\omega_m|\leq |v|^\beta$ in $\mathbb{R}^N$ and
$|\omega_m|= |v|^\beta$ in $A_m$, we have
\[
\Big[ \int_{A_m }|v|^{r\beta}\Big]^{2/r}\leq S\beta^2
\|v\|_r^{q-2} \Big[\int_{\mathbb{R}^N}|v|^{2q_1\beta} \,dx
\Big]^{1/q_1}.
\]
By the Monotone Convergence Theorem, we obtain
\begin{equation}
\|v\|_{r\beta} \leq \beta^{1/\beta}
(S \|v\|_r^{q-2})^{1/(2\beta)}\|v\|_{2\beta q_1}. \label{importante}
\end{equation}
Since $q< r$,  we have  $r>2q_1$. Set $\sigma = r/{2q_1}>1$.
Setting  $\beta = \sigma$ in \eqref{importante}, we obtain
$2q_1\beta=r$ and
\begin{equation}
\|v\|_{r\sigma} \leq \sigma^{1/\sigma}(S \|v\|_r^{q-2})
^{1/(2\sigma)}\|v\|_{r}. \label{loop1}
\end{equation}
Taking  $\beta=\sigma^2$ in \eqref{importante}, we obtain
$2q_1\beta=r\sigma$ and
\begin{equation}
\|v\|_{r\sigma^2} \leq \sigma^{2/\sigma^2}
(S \|v\|_r^{q-2})^{ 1/(2\sigma^2)}\|v\|_{r\sigma}. \label{loop2}
\end{equation}
 From \eqref{loop1} and \eqref{loop2}, we find
\begin{equation}
\|v\|_{r\sigma^2} \leq \sigma^{\frac 1\sigma+\frac 2{\sigma^2}}(S
\|v\|_r^{q-2})^{\frac 12(\frac 1{\sigma}+\frac
1{\sigma^2})}\|v\|_{r}. \label{passo2}
\end{equation}
The result is obtained by iteration of the estimate \eqref{importante}.
Taking $\beta = \sigma^j$, $j=1,2, \dots$,  yields
\begin{equation}
\|v\|_{r\sigma^m} \leq \sigma^{\frac 1\sigma+\frac 2{\sigma^2}+\frac
3{\sigma^3}+...+\frac j{\sigma^j}}(S \|v\|_r^{q-2})^{\frac 12(\frac
1{\sigma}+\frac 1{\sigma^2}+\frac 1{\sigma^3}+...+\frac
1{\sigma^j})}\|v\|_{r}. \label{passoM}
\end{equation}
Since the series bellow are convergent and
\[
 \sum_{j=1}^\infty \frac j{\sigma^j}= \frac {\sigma}{(\sigma-1)^2},
\quad
\frac 12\sum_{j=1}^\infty \frac 1{\sigma^j}=\frac 1{2(\sigma-1)},
\]
from \eqref{passoM}, we have
\[
\|v\|_p \leq \sigma^{\sigma/(\sigma-1)^2}
(S \|v\|_r^{q-2})^{\frac 1{2(\sigma-1)}}\|v\|_{r},
\]
for all $p\geq r$.  Since  $b(x)$ is nonnegative,
we have $\|v\|_r\leq S^{1/2}C^{1/2}$.  Using
that
\[
\|v\|_\infty =\lim_{p\to +\infty} \|v\|_p,
\]
we conclude  that Proposition \ref{prop1} is valid for
\[
k = \sigma^{\frac {\sigma}{(\sigma-1)^2}}(S^{q/2} C^{(q-2)/2})^{\frac
1{2(\sigma-1)}}S^{1/2}C^{1/2}.
\]
\end{proof}

\section{Auxiliary problem}\label{section3}

In this section we study the existence of a solution for the
 Schr\"odinger-Poisson  system with subcritical growth.
This result will be  useful for obtaining our main result.
More precisely, we consider the system
\begin{equation} \label{eAP}
\begin{gathered} - \Delta u +V(x)u+\phi u= h(u),  \quad
\text{in } \mathbb{R}^3, \\
-\Delta \phi=u^2, \quad\text{in } \mathbb{R}^3,
\end{gathered}
\end{equation}
where $V:\mathbb{R}^3 \to \mathbb{R}$ is a bounded locally H\"older
continuous that satisfies (V0) and (V1), and the function
$h\in C(\mathbb{R}^+,\mathbb{R})$ and satisfies:
\begin{itemize}
\item[(H1)] $h(0)=0$;

\item[(H2)] $\lim_{s\to 0 ^+} h(s)/s = 0$;

\item[(H3)] There exist $C>0$ and $p \in (4,6)$ such that
$$
|h(s)|\leq C(|s|+|s|^{p-1}), \forall s \in \mathbb{R}^+;
$$

\item[(H4)] $\lim_{s\to +\infty} H(s)/s^4=+\infty$,
where $H(s)=\int_0^s h(t)dt$;

\item[(H5)] For $\alpha >0$ given by (V0), there exists
$\sigma \in (0, \alpha)$ such that
$$
sh(s)-4H(s)\geq -\sigma s^2
$$
for all $s \neq 0$.
\end{itemize}

Next we  review some of the standard facts on the
Schr\"odinger-Poisson equations (see \cite{ap,G,Ruiz,zz}).
We begin by observing  that    \eqref{eAP} can be transformed
into a Schr\"odinger equation with a nonlocal term. In fact,
 by Lax-Milgram theorem, given  $u\in
H^1(\mathbb{R}^3)$, there exists a unique
$\phi=\phi_u \in {D}^{1,2}(\mathbb{R}^3)$ such that
$$
-\Delta \phi=u^2.
$$
The function  $\phi_u$ has the following properties (see
\cite{Daprile1}):

\begin{lemma}\label{lm1}
\begin{itemize}
\item[(i)]
 There exists $C>0$ such that
$\|\phi_u\|_{{D}^{1,2}(\mathbb{R}^3)}\leq C\|u\|^2$ and
$$
\int_{\mathbb{R}^3}|\nabla \phi_u|^2\,dx=\int_{\mathbb{R}^3}\phi_u
u^2\,dx\leq C \|u\|^4,\quad  \forall u\in H^1(\mathbb{R}^3);
$$

\item[(ii)] $\phi_u\geq 0$, $\forall u\in H^1(\mathbb{R}^3)$;

\item[(iii)]
 $\phi_{tu}=t^2\phi_u$, $\forall t>0, u\in H^1(\mathbb{R}^3)$.

\item[(iv)] If $y\in \mathbb{R}^3$ and $\tilde u(x)=u(x+y)$, then
$\phi_{\tilde u}(x) = \phi_{u} (x+y)$ and
$$
\int_{\mathbb{R}^3}\phi_{\tilde u} \tilde u ^2\,dx
=\int_{\mathbb{R}^3}\phi_u u^2\,dx;
$$

\item[(v)] if $u_n \rightharpoonup u$ in $H^1(\mathbb{R}^3)$, then
$\phi_{u_n} \rightharpoonup \phi_u$ in ${D}^{1,2}(\mathbb{R}^3)$.
\end{itemize}
\end{lemma}


 From (V0), we can see that the
$H^1(\mathbb{R}^3)$ norm is equivalent to
$$
\|u\|^2=\int_{\mathbb{R}^3}(|\nabla u|^2+V(x)u^2)\,dx.
$$
Let $I$ be the functional on $H^1(\mathbb{R}^3)$ defined by
\begin{equation}\label{defI}
I(u) = \frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V(x)u^2)\,dx
+ \frac14\int_{\mathbb{R}^3} \phi_uu^2\,dx
- \int_{\mathbb{R}^3}H(u)\,dx.
\end{equation}
 From the conditions on $h$, the functional
$I\in C^1(H^1(\mathbb{R}^3),\mathbb{R})$ and its Gateaux derivative is
$$
I'(u)v= \int_{\mathbb{R}^3} (\nabla u \cdot \nabla v + V(x) uv)\,dx
+\int_{\mathbb{R}^3}\phi_u u v\,dx - \int_{\mathbb{R}^3}h(u)v\,dx,
$$
for every $u, v \in H^1(\mathbb{R}^3)$. Hence, corresponding to
each critical point  of $I$ there is a weak solution of the
Schr\"odinger equation with a nonlocal term:
\begin{equation}
-\Delta u +V(x)u+\phi_u u= h(u),\quad \text{in } \mathbb{R}^3.
\label{PP}
\end{equation}



\begin{lemma} \label{lm2}
Suppose that  $V$ satisfies {\rm (V0)} and $h$
satisfies {\rm (H1)--(H5)}. If $(u_n)\subset H^1(\mathbb{R}^3)$ is a
Cerami sequence of $I$; i. e., $(I(u_n))$ is bounded and
$(1+\|u_n\|)I'(u_n)\to 0$,  then $(u_n)$ is bounded in
$H^1(\mathbb{R}^3)$.
\end{lemma}

\begin{proof}
 From (H5),
\begin{align*}
4I(u_n)-I'(u_n)(u_n)
&=  \|u_n\|^2+ \int_{\mathbb{R}^3}[(u_n)h(u_n)-4H(u_n)]\,dx\\
&\geq  \|u_n\|^2-\sigma \int_{\mathbb{R}^3}u_n^2\,dx
\geq \big( 1-\frac \sigma \alpha \big)\|u_n\|^2.
\end{align*}
Since  $(4I(u_n)-I'(u_n)(u_n))$ is bounded, we conclude that
$(u_n)$ is bounded in $H^1(\mathbb{R}^3)$.
\end{proof}

\begin{lemma}\label{lm3}
Suppose that  $V$ satisfies {\rm (V0)}   and  $h$  satisfies
{\rm (H1)--(H4)}. Then, there exist $\rho>0$ and
$e\in H^1(\mathbb{R}^3)$ with $\|e\|>\rho$, such that
$$
b\doteq\inf_{\|u\|=\rho}I(u)>I(0)=0\geq I(e).
$$
\end{lemma}

\begin{proof}
 From (H2)--(H3), for each $\epsilon >0$
there exists $C_\epsilon >0$ such that
\[
H(s) \leq \epsilon s^2 + C_\epsilon s^p, \quad \forall\, s \in
\mathbb{R}.
\]
By Sobolev inequalities, there exist positive constants $\alpha$ and
$\beta$ such that
\[
I(u) \geq \big[(\frac12 - \epsilon \alpha) - \beta C_\epsilon
\|u\|^{p-2}\big]\|u\|^2
\]
We can assume, by decreasing $\epsilon$ if necessary, that there
exist positive numbers $b, \rho$ such that
$b = \inf\{I(u),\|u\|=\rho\} > I(0)=0$.

  From (H4), for any $v\in
H^1(\mathbb{R}^3)$ and $M >(1/4)\int_{\mathbb{R}^3}\phi_vv^2 \,dx$,
there exists $C>0$ such that $ H(s) \geq Ms^4 - Cs^2$, for all
$s\in \mathbb{R}$. Hence,
\[
I(tv) \leq (C+\frac12)\|v\|^2t^2 -
\Big(M-\frac14\int_{\mathbb{R}^3}\phi_vv^2 \,dx\Big)t^4 \to
-\infty,\quad \text{as }t\to \infty.
\]
Thus, $e=tv$ satisfies  $\|e\| >\rho$
and $I(e) <0 = I(0)$, provided  $t$ sufficiently large.
\end{proof}

 By a version of the mountain pass theorem (see \cite{Ek}),
there is a Cerami sequence $(u_n)\subset H^1(\mathbb{R}^3)$ such
that
$$
I(u_n)\to c\quad  \text{and} \quad  (1+\|u_n\|)I'(u_n)\to 0,
$$
where
$$
c= \inf_{\gamma \in \Gamma}\max_{t\in [0,1]} I(\gamma(t)), \quad
\Gamma = \{ \gamma:[0,1]\to H^1(\mathbb{R}^3): \gamma
(0)=0, \gamma(1)=e \}.
$$

The main result of this section is the following.

\begin{proposition}\label{prop2}
Suppose that $V$ satisfies {\rm (V0), (V1)}, and  $h$  satisfies
{\rm (H1)--(H5)}. Then  \eqref{eAP} possesses a
positive solution $u$ such that
 $\|u\|^2\leq 4c\alpha/(\alpha-\sigma)$, where $\alpha$
and $\sigma$ are given by {\rm(V0)} and {\rm (H4)} respectively
and  $c$ is the minimax level associated with  \eqref{eAP}.
\end{proposition}

 \begin{proof}
 By Lemma \ref{lm2}, we can assume that  $(u_n)$ is weakly
convergent to $u$, for some  $u\in H^1(\mathbb{R}^3)$.
Taking $v\in C^\infty_0(\mathbb{R}^3)$, from Lemma \ref{lm1}(v),
$\phi_{u_n} \rightharpoonup
\phi_u$ in ${D}^{1,2}(\mathbb{R}^3)$, as $n\to \infty$, and so
\[
\int_{\mathbb{R}^3}\phi_{u_n}u v\,dx \to  \int_{\mathbb{R}^3}\phi_{u}u
v\,dx,\quad \text{as }n\to \infty.
\]
Moreover, using H\"{o}lder's inequality we obtain
\[
|\int_{\mathbb{R}^3}\phi_{u_n} (u_n-u) v\,dx| \leq
\|\phi_{u_n}\|_{L^{2^*}(\mathbb{R}^3)}\|u_n - u\|_{L^{12/5}(\Omega)
}\|v\|_{L^{12/5}(\Omega)}=o_n(1),
\]
where $\Omega= \operatorname{supp}v$. Therefore,
$$
\int_{\mathbb{R}^3}\phi_{u_n} u_n v\,dx-\int_{\mathbb{R}^3}\phi_{u} u
v\,dx= \int_{\mathbb{R}^3}(\phi_{u_n}-\phi_u) u v\,dx +\int_{\mathbb{R}^3}\phi_{u_n} (u_n-u) v\,dx= o_n(1),
$$
for all $v\in C^\infty_0(\mathbb{R}^3)$,  which implies
 $$
I'(u)v=0,  \quad \text{for all } v \in H^1(\mathbb{R}^3).
$$
Consequently, $u$ is a weak solution for \eqref{PP}. To
conclude the proof, it only remains to show that $u\neq 0$.
Assume by contradiction that  $u\equiv 0$. By \cite[Lemma 2.1]{zr}
(see also \cite{L1}), we can claim that only one of the following
conditions hold:
\begin{itemize}
\item[(i)] For all $q\in (2,2^*)$,
$$
\lim_{n\to +\infty} \int_{\mathbb{R}^3}|u_n|^q\,dx=0.
$$

\item[(ii)] There are positive numbers $R$ and $\eta$,
and a sequence $(y_n)\subset \mathbb{R}^3$ such that
$$
\liminf_{n\to +\infty} \int_{B_R(y_n)}u_n^2\,dx > \eta>0.
$$
\end{itemize}
If $(i)$ occurs, then from (F2) and  (F3), we have
$$
\lim_{n\to +\infty} \int_{\mathbb{R}^3}h(u_n)u_n\,dx=0.
$$
By Lemma \ref{lm1}(ii),
$$
\|u_n\|^2\leq\|u_n\|^2+\int_{\mathbb{R}^3}\phi_{u_n} u_n^2\,dx =
\int_{\mathbb{R}^3}h(u_n)u_n\,dx+o_n(1).
$$
As a consequence, the sequence $(u_n)$ is strongly convergent in
$H^1(\mathbb{R}^3)$ to $0$. Then $I(u_n)\to 0$, contrary to
$I(u_n)\to c>0$. Hence, (ii) is valid.   From (V1) we
can assume that $y_n \in \mathbb{Z}^N$. Define
$$
 \tilde{u}_n(x)=u_n(x+y_n).
$$
 From (V1) again, $(\tilde{u}_n)$ is bounded in $H^1(\mathbb{R}^3)$
and we can clearly assume that $(\tilde{u}_n)$ is weakly convergent
to $\tilde{u}$ for some $\tilde{u}\in H^1(\mathbb{R}^3)$.  From
(ii), $\tilde{u}\neq 0$. Observing that  Lemma  \ref{lm1}(iv)
implies that
\[
I'(\tilde{u}_n)\tilde{u}_n = I'({u}_n)u_n\quad \text{and}\quad
I(\tilde{u}_n) = I({u}_n),
\]
hence that  $(\tilde{u}_n)$ is a Cerami sequence of $I$, and finally
$$
I'(\tilde{u})=0\quad  \mbox {with}\quad  \tilde{u}\neq 0,
$$
where we have again used Lemma \ref{lm1}(v). It follows that
$\tilde{u}$ is a nontrivial solution to  \eqref{PP}.
 Using bootstrap arguments and the maximum principle,
we can conclude that the solution $\tilde{u}$ is positive.

Finally,  to verify that $\tilde{u}$ satisfies inequality
$\|u\|^2\leq 4c\alpha/(\alpha-\sigma)$, we observe that from
(H5),
\begin{align*}
4I(\tilde u_n) - I'(\tilde u_n)\tilde u_n \geq \big( 1-\frac
\sigma \alpha \big) \|\tilde u_n\|^2, \quad \forall n.
\end{align*}
Passing to the limit we obtain
\begin{align*}
4c &=  \liminf_{n\to \infty} (4I(\tilde u_n) - I'(\tilde u_n)\tilde
u_n) \geq \big( 1-\frac \sigma \alpha \big) \|u\|^2,
\end{align*}
and the proof is complete.
\end{proof}

\section{Preliminary results}

To establish the existence of a solution to \eqref{eP},  we define
a sequence of functions $\{g_n\}$ by setting
$$
g_n(s)= \begin{cases}
0, & \text{if }s \leq 0 \\
g(s), & \text{if } 0 \leq s \leq M_n \\
\frac{g(M_n)}{M_n^{q-1}}s^{q-1}, & \text{if } s \geq M_n.
\end{cases}
$$
 From  (F6), we have
\begin{equation}
|g_n(s)| \leq \frac{g(M_n)}{M_n^{q-1}}|s|^{{q-1}} \quad
 \text{for all } s.\label{hmgmp-q}
\end{equation}
We conclude from  (F3) that  $f_{\lambda,n}(s)=f_o(s)+ \lambda g_n(s)$
satisfies
\begin{equation}\label{eq1.1}
|f_{\lambda,n}(s)| \leq (1+\lambda g(M_n)M_n)|s|^{q-1},
\end{equation}
which implies that the problem
\begin{equation} \label{Plambdan}
\begin{gathered}
- \Delta u +V(x)u+\phi u= f_{\lambda,n}(u), \quad\text{in }
  \mathbb{R}^3, \\
-\Delta \phi=u^2, \quad\text{in } \mathbb{R}^3,
\end{gathered}
\end{equation}
is variational for every  $\lambda >0$ and  $n\in \mathbb{N}$.
The  functional associated with  \eqref{Plambdan} is denoted by
$J_{\lambda,n}:H^1(\mathbb{R}^3)\times D^{1,2}(\mathbb{R}^3)
\to \mathbb{R}$  and given by
\begin{align*}
&J_{\lambda,n}(u,\phi) \\
&= \frac 12 \int_{\mathbb{R}^3}(|\nabla
u|^2+V(x)u^2)\,dx -\frac 14\int_{\mathbb{R}^3}|\nabla \phi|^2\,dx
+ \frac 12\int_{\mathbb{R}^3}\phi u^2\,dx
- \int_{\mathbb{R}^3}F_{\lambda,n}(u)\,dx.
\end{align*}
We observe that $J_{\lambda,n}$ is strongly indefinite. To
overcome this  difficulty, we introduce the functional
$I_{\lambda,n}: H^1(\mathbb{R}^3)\to \mathbb{R}$ defined by
$$
I_{\lambda,n}(u)=\frac 12 \int_{\mathbb{R}^3}(|\nabla u|^2+V(x)u^2)\,dx
+\frac 14\int_{\mathbb{R}^3}\phi_u u^2\,dx
- \int_{\mathbb{R}^3}F_{\lambda,n}(u)\,dx,
$$
with $\phi_u$ being the function defined in Section \ref{section3}.
By \eqref{eq1.1}, the functional
$I_{\lambda,n}\in C^1(H^1(\mathbb{R}^3),\mathbb{R})$ and  its Gateaux
derivative is
$$
I_{\lambda,n}'(u)v= \int_{\mathbb{R}^3} (\nabla u \nabla v + V(x)
uv)\,dx +\int_{\mathbb{R}^3}\phi_u u v\,dx
- \int_{\mathbb{R}^3}f_{\lambda,n}(u)v\,dx,
$$
for every $u,v \in H^1(\mathbb{R}^3)$. Hence,  corresponding
to each  critical point of  $I_{\lambda,n}$
there exists a weak solutions of
\begin{equation} \label{P'lambdan}
\begin{gathered}
- \Delta u +V(x)u+\phi_u u= f_{\lambda,n}(u), \quad\text{in }
 \mathbb{R}^3, \\
 \quad u\in H^1(\mathbb{R}^3).
\end{gathered}
\end{equation}
For $F_o$ given by (F4), we introduce an auxiliary Euler-Lagrange
functional $I_o: H^1(\mathbb{R}^3) \to \mathbb{R}$ given by
$$
I_o(u)=\frac 12 \|u\|^2 +\frac 14\int_{\mathbb{R}^3}\phi_u u^2\,dx-
\int_{\mathbb{R}^3}F_o(u)\,dx.
$$
 From (F1)--(F4) and Lemma \ref{lm1}(i,iii), it is standard to
check that $I_o$ possesses the geometric hypotheses of the
 mountain pass theorem  (see Lemma \ref{lm3}).
 Then, there exist $e \in H^1(\mathbb{R}^3)$ and $c_o \in \mathbb{R}$
such that
$$
c_o= \inf_{\gamma \in \Gamma}\max_{t\in [0,1]} I_o(\gamma(t)) >0,
$$
where
\begin{equation}
\Gamma = \{ \gamma\in C([0,1], H^1(\mathbb{R}^3)) : \gamma (0)=0,
\gamma(1)=e \} \neq \emptyset. \label{gama}
\end{equation}
Since  $f_{\lambda,n}$ satisfies
conditions (H1)--(H5) of Proposition \ref{prop2}, for every
$\lambda >0$ and $n\in \mathbb{N}$, and $V$ satisfies (V0)--(V1),
the problem \eqref{P'lambdan} has a positive solution such that
$u_{\lambda,n}\in H^1(\mathbb{R}^3)$ and
$$
\|u_{\lambda,n}\|^2\leq c_{\lambda,n}\alpha/(\alpha-\sigma)
$$
where $c_{\lambda,n}= \inf_{\gamma \in \Gamma}\max_{t\in [0,1]}
I_{\lambda,n}(\gamma(t))$  and  $\Gamma$ is defined by \eqref{gama}
and is independent of $\lambda$ and $n$. In fact, since
$g(s)\geq 0$ for all $s$, we have $F_{\lambda,n}(s)\geq F_o(s)$.
Hence
\begin{equation}
I_{\lambda,n}(v)\leq I_o(v), \quad\text{for all }
v\in H^1(\mathbb{R}^3). \label{I0}
\end{equation}
In particular, $I_{\lambda,n}(e)\leq I_o(e)<0$. Thus, $\Gamma$
is independent of $\lambda$ and $n$. Moreover, from \eqref{I0},
we have
\begin{equation}
c_{\lambda,n} \leq c_o. \label{c<co}
\end{equation}

\section{Proof of Theorem \ref{thm1}}

Our proof consists in finding $n$, $\lambda$ and a positive solution
$u$ of  \eqref{Plambdan} such that $u(x)\leq M_n$, for all
$x \in \mathbb{R}^3$. It is immediate that  $u$ and
$\phi=\phi_u$ solve  problem \eqref{eP}.

For $k=k(q,4c_o\alpha/(\alpha-\sigma))$ given by Proposition
\ref{prop1},   we fix $n$ such that $M_n^2>k$.  Let
$\lambda_o >0$ be such that $\lambda_o g(M_n)M_n \leq 1$.  From
\eqref{eq1.1},  we have
$$
|f_{\lambda,n}(s)|\leq 2|s|^{q-1},\quad  \forall s.
$$
By Proposition \ref{prop2}, there exists a solution
$u=u_{\lambda,n}$ of \eqref{Plambdan} such that
$\|u\|^2\leq 4c_{\lambda,n}\alpha/(\alpha-\sigma)$.
Combining this inequality with  \eqref{c<co}, yields
$$
 \|u\|^2\leq 4c_{\lambda,n}\alpha/(\alpha-\sigma)
\leq 4c_o\alpha/(\alpha-\sigma).
$$
Invoking Lemma \ref{lm1}(ii),  we conclude that $\phi_{u}\geq 0$
and consequently $V(x)+\phi_{u}$
is a non-negative function. Using Proposition \ref{prop1}, with
$b(x)=V(x)+\phi_{u}$, we obtain that $\|u\|_\infty \leq K$, for some
$K=K(q,c_o)>0$, and the proof is complete.


\subsection*{Acknowledgments}

This work was done while M. A. S. Souto  was visiting the
Universidade de S\~{a}o Paulo at S\~{a}o Carlos.
He thanks all the faculty and staff of the Department of
Mathematics for their support and kind hospitality.

We are  grateful to the anonymous referee for a number of helpful
comments that improved this article and also to Professor Alfonso Castro for your attention with us.

\begin{thebibliography}{00}

\bibitem{as}  {Alves, C. O.};   {Souto, M. A. S.};
     {On existence of solution for a class of semilinear elliptic equations wit nonlinearities that lies between two different powers},
     {Abst. and Appl. Analysis}  ({2008}),    {1--7}.

\bibitem{AM}{Ambrosetti, A.}; { Ruiz, D.};
    {Multiple bound states for the {S}chr\"odinger-{P}oisson  problem},
    {Commun. Contemp. Math.} {10} {(2008)}, {391--404}.

\bibitem{ap}  {Azzollini, A.};     {Pomponio, A.};
 Ground state solutions for the nonlinear Schr\"{o}dinger-Maxwell
equations, J. Math. Anal. Appl. 345 (2008), 90--108.

\bibitem{bf}  {Benci, V.};     {Fortunato, D.};
     {An eigenvalue problem for the Schr\"odinger-Maxwell equations},
     {Top. Meth. Nonlinear Anal.}
     {11}     ({1998}),     {283--293}.

\bibitem{Cerami} {Cerami, G.}; {Vaira, G.};
 {Positive solutions for some non-autonomous {S}chr\"odinger-{P}oisson
 systems},   {J. Differential Equations} {248} {(2010)}, {521--543}.

\bibitem{C}{Coclite, G. M.};
{A multiplicity result for the nonlinear
              {S}chr\"odinger-{M}axwell equations},
{Commun. Appl. Anal.} {7} {(2003)}, {417--423}.

\bibitem{zr}  {Coti-Zelati, V.}; {Rabinowitz, P. H.};
 {Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^3$},
{Comm. Pure and Appl. Math} {45} {(1992)},     {1217--1269}.


\bibitem{Daprile1} {D'Aprile, T.}; {Mugnai, D.};
 {Solitary waves for nonlinear Klein-Gordon-Maxwell and
              Schr\"o\-dinger-Maxwell equations},
{Proc. Roy. Soc. Edinburgh Sect. A} {134}
 {(2004)}, {893--906}

\bibitem{Daprile2} {D'Aprile, T.}; {Mugnai, D.};
 {Non-existence results for the coupled
{K}lein-{G}ordon-{M}axwell  equations}, {Adv. Nonlinear Stud.} {4}
{(2004)}, {307--322}.

\bibitem{Davenia} {d'Avenia, P.};
    {Non-radially symmetric solutions of nonlinear {S}chr\"odinger
              equation coupled with {M}axwell equations},
 {Adv. Nonlinear Stud.} {2} {(2002)}, {177--192}.


\bibitem{Ek}  {Ekeland, I.};
 {Convexity Methods in Hamilton Mechanics},
{Springer Verlag},  {1990}.

\bibitem{G} {Gaetano, S.};
  {Multiple positive solutions for a  Schr\"odinger-Poisson-Slater system},
     {J. Math. Analysis and Appl.}      {365},    ({2010})     {288--299}.

\bibitem{Ianni}  {Ianni, I.}; {Vaira, G.};
 {On concentration of positive bound states for the
              {S}chr\"odinger-{P}oisson problem with potentials},
{Adv. Nonlinear Stud.} {8} {(2008)}, {573--595}.

\bibitem{K1}{Kikuchi, H.};
 {On the existence of a solution for elliptic system related to
              the {M}axwell-{S}chr\"odinger equations},
{Nonlinear Anal.} {67} {(2007)}, {1445--1456}.

\bibitem{K2}{Kikuchi, H.};
{Existence of standing waves for the nonlinear {S}chr\"odinger
              equation with double power nonlinearity and harmonic
              potential},
{Asymptotic analysis and singularities---elliptic and parabolic
              {PDE}s and related problems},
 {Adv. Stud. Pure Math.}  {47}
{623--633}, {Math. Soc. Japan}, {Tokyo}, {(2007)}.

\bibitem{L1}  {Lions, P. L.};
 {The concentration-compactness principle in the calculus of variations. The locally compact case, part 2},
{Analyse Nonlin\'{e}aire}, {I} {(1984)},  {223--283}.


\bibitem{Ruiz} {Ruiz, D.};
     {The Schr\"odinger-Poisson equation under the effect of a nonlinear local term},
     {J. Funct. Analysis}      {237}    ({2006}),     {655--674}.

\bibitem{Ruiz2} {Ruiz, D.}; { Gaetano, S.};
 {A note on the {S}chr\"odinger-{P}oisson-{S}later equation on
              bounded domains},
{Adv. Nonlinear Stud.} {8} {(2008)}, {179--190}.

\bibitem{zz} {Zhao, F.}; {Zhao, L.};
 Positive solutions for Schr\"{o}dinger-Poisson
equations with a critical exponent, Nonlinear Anal. 70 (2009),
2150--2164.

\end{thebibliography}

\end{document}