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\markboth{Stationary Solutions for a Schr\"odinger-Poisson System}
{ Khalid Benmlih }

\begin{document}
\setcounter{page}{65}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
2002-Fez conference on Partial Differential Equations,\newline
Electronic Journal of Differential Equations, Conference 09, 2002,
pp 65--76. \newline http://ejde.math.swt.edu or
http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
  Stationary Solutions for a Schr\"odinger-Poisson System in $\mathbb{R}^3$
%
\thanks{ {\em Mathematics Subject Classifications:} 35J50, 35Q40.
\hfil\break\indent {\em Key words:} Schr\"odinger equation,
Poisson equation, standing wave solutions, \hfil\break\indent
variational methods.
\hfil\break\indent \copyright 2002 Southwest Texas State
University. \hfil\break\indent Published December 28, 2002.} }

\date{}
\author{Khalid Benmlih}
\maketitle

\begin{abstract}
  Under appropriate, almost optimal, assumptions on the data we prove
  existence of standing wave solutions for a nonlinear Schr\"odinger
  equation in the entire space $\mathbb{R}^3$ when the real electric
  potential satisfies a linear Poisson equation.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}


\section{Introduction}

Consider the time-dependent system which couples the Schr\"odinger
equation
\begin{equation}
i \partial_t u =  -\frac 12 \Delta u +  ( V + \widetilde V ) u
\label{S}
\end{equation}
with initial value $ u(x,0)  =  u(x)$, and the Poisson equation
\begin{equation}
-\Delta  V  = | u |^2 -n^* .\label{P}
\end{equation}
The dopant-density $n^*$ and the effective potential $\widetilde V
$ are given time-independent reals functions. There are many
papers dealing with the physical problem modelled by this system
from which we mention  Markowich, Ringhofer \& Schmeiser [8];
Illner, Kavian \& Lange [3]; Nier [9]; Illner, Lange, Toomire \&
Zweifel [4], and references therein.

In this work we are mainly concerned with the proof of standing
waves (actually ground states)  of \eqref{S}--\eqref{P} in the
entire space $\mathbb{R}^3$, i.e. solutions of the form
$$ u(x,t) = e^{i \omega t}  u (x)
$$
with real number $\omega$ (frequency) and real wave function $u$.
Hence we are interested in the stationary system
\begin{gather}
-\frac 12 \Delta u  +  (V + \widetilde V )u   +  \omega  u
= 0 \quad \mbox{in }   \mathbb{R}^3 \label{1.1}   \\
-\Delta V = | u |^2  -  n^*  \quad \mbox{in }  \mathbb{R}^3
\label{1.2}
\end{gather}
under appropriate, almost optimal, assumptions on $\widetilde V$
and $n^*$. We suppose first that $ \widetilde V \in L^1_{\rm
loc}(\mathbb{R}^3) $ and $ n^*\in L^{6/5}(\mathbb{R}^3)$.

Let us remark that if $V_0$ is such that $- \Delta V_0 = -  n^*$
then $(0, V_0)$ is a solution of  the system
\eqref{1.1}-\eqref{1.2}. But here, we deal with solutions $(u, V)$
in $H^1 (\mathbb{R}^3) \times {\cal D}^{1,2} (\mathbb{R}^3)$ such
that $u\not\equiv 0$.

F. Nier [9] has studied the system \eqref{1.1}-\eqref{1.2}. He has
showed the existence of a solution for small data i.e. when $
\|\widetilde V \|_{L^2}$ and $ \|n^* \|_{L^2}$ are small enough.
Conversely to our approach here, he has began by solving
\eqref{1.1} for a fixed $V$ and investigate the Poisson equation
then obtained.

In this paper we solve first explicitly the Poisson equation
\eqref{1.2} for a fixed $u$ in $H^1 (\mathbb{R}^3)$. Next we
substitute this solution $V = V(u)$ in the Schr\"odinger equation
\eqref{1.1} and look into the solvability of
\begin{equation}
-\frac 12 \Delta u  +  ( V(u) + \widetilde V )  u   +  \omega u =
0 \quad \mbox{in }  \mathbb{R}^3  .\label{1.3}
\end{equation}
Using the explicit formula of $V(u)$, this equation appears as a
{\sl Hartree equation} studied by P.L. Lions [6] in the case where
$n^* \equiv 0$ and $\widetilde V (x):= -2/|x|$. The fact that
$\widetilde V$ in [6] converges to zero at infinity plays a
crucial role to prove existence of solutions. However, in this
paper we show that a slight modification of the arguments used in
that paper allows us to prove existence of a ground state in the
case $\widetilde V$ satisfying \eqref{1.5}, \eqref{1.7} and $n^*$
not necessarily zero (but satisfying \eqref{1.6} and \eqref{1.7}
as below).

Before giving our hypotheses on $\widetilde V$ and $n^*$ let us
define a decomposition which will be useful in the sequel.

\begin{definition} \label{def1.1} \rm
We say that $g$ satisfies the decomposition \eqref{1.4} if:
\begin{itemize}
\item[(i)] $g \in L^1_{\rm loc} (\mathbb{R}^3)$,
\item[(ii)] $g \geq 0$, and
\item[(iii)] There exists $q_0 \in [3/2 , \infty ]$ :
$\forall \lambda > 0$  $\exists g_{1\lambda} \in
L^{q_0}(\mathbb{R}^3) , q_{\lambda} \in ]3/2 , \infty [$ and
$g_{2\lambda} \in L^{q_\lambda} (\mathbb{R}^3)$ such that
\begin{equation}
g = g_{1\lambda} + g_{2\lambda} \quad \mbox{and} \quad
\lim_{\lambda \to 0} \|g_{1\lambda} \|_{L^{q_0}} = 0.\label{1.4}
\end{equation}
\end{itemize}
\end{definition}

\noindent For convenience, we use throughout this paper the following
notations:
\begin{itemize}
\item $\|.\|$ denotes the norm $\|.\|_{L^2}$ on $L^2 (\mathbb{R}^3)$,
\item $\mathbb{I}_A$ denotes the characteristic function of the set
$A\subset \mathbb{R}^3$,
\item $[ F \leq \lambda ]$ denotes the  set $\{ x ;   F(x) \leq
\lambda \}$ for a function $F$ and $\lambda \in \mathbb{R} $.
\end{itemize}
Let us give now two examples of functions satisfying the
conditions in Definition \ref{def1.1}.

\begin{example} \label{ex1.2} \rm
The following two functions satisfy the decomposition \eqref{1.4}:
\begin{itemize}
\item[(i)] $g(x):=1/|x|^\alpha$  for some $ 0< \alpha < 2$.
\item[(ii)] $|g|$ where $g$ is a function in $L^r (\mathbb{R}^3)$
for some $ r > 3/2$.
\end{itemize}
\end{example}

\paragraph{Proof.}  To prove (i) we write, for $\lambda > 0$,
$$\frac 1{|x|^\alpha}
:= \underbrace{\frac 1{|x|^\alpha} \mathbb{I}_{[|x|>
1/\lambda]}}_{g_{1\lambda}} +  \underbrace{\frac 1{|x|^\alpha}
\mathbb{I}_{[|x|\leq 1/\lambda]}}_{g_{2\lambda}}.
$$
Elementary calculations give
$$\|g_{1\lambda}\|_{L^{q_0}}^{q_0}= \frac{4\pi}{\alpha q_0 - 3}
( \lambda )^{\alpha q_0 - 3}\quad {\rm and}\quad \|
g_{2\lambda}\|_{L^{q}}^{q}= \frac{4\pi}{3- \alpha q} ( \frac
1{\lambda} )^{3- \alpha q}.
$$
Hence it suffices to choose any finite numbers $q_0$ , $q$ such
that $3/2 < q < 3/\alpha < q_0$.

\noindent To show (ii) write, as above,
$$
|g|:= \underbrace{|g| \mathbb{I}_{[|g|\leq
\lambda]}}_{g_{1\lambda}} +  \underbrace{|g| \mathbb{I}_{[|g| >
\lambda]}}_{g_{2\lambda}} .
$$
It is clear that $ \| g_{1\lambda} \|_{L^\infty} \leq \lambda $
($q_0=\infty$) and $\| g_{2\lambda} \|_{L^r} \leq \|g \|_{L^r}$
($q_\lambda =r$).  \hfill$\square$

\paragraph{Hypotheses.} In what follows we assume that
\begin{equation}
\widetilde V^+ \in L^1_{\rm loc}(\mathbb{R}^3) \quad\mbox{and}\quad
\widetilde V^- \mbox{ satisfies the decomposition \eqref{1.4} },
\label{1.5}
\end{equation}
where ${\widetilde V}^+ (x) := \max (\widetilde V (x) , 0 )$ and 
${\widetilde V}^- (x) := \max (-\widetilde V (x) , 0 )$. We
suppose also that
\begin{equation}
n^* \in L^1 \cap L^{6/5} (\mathbb{R}^3) \label{1.6}
\end{equation}
and finally if we denote by
$$\varrho (x):= 2 \widetilde
V (x) -   \frac{1}{2\pi} \int_{\mathbb{R}^3} \frac{n^*(y)}{|x-y|}
\,dy
$$
we assume that
\begin{equation}
\inf \Big\{\int_{\mathbb{R}^3} \big(|\nabla \varphi |^2+ \varrho
(x) \varphi^2\big)dx ,\int_{\mathbb{R}^3} |\varphi |^2= 1 \Big\} <
0. \label{1.7}
\end{equation}
Remark that in the case of [6] (where $n^* \equiv 0$ and
$\widetilde V (x):= -2/|x|$), all the three hypotheses above are
satisfied. Indeed, \eqref{1.5} and \eqref{1.6} follow from $(i)$
of Example \ref{ex1.2}. Moreover, if we consider $\Phi (x):=
e^{-2 |x|}$ then it verifies
$$ -\Delta \Phi - 4 {\Phi \over | x | } = -4 \Phi ,
$$
and consequently
$$
\inf \Big\{\int_{\mathbb{R}^3} |\nabla \varphi |^2-4
\int_{\mathbb{R}^3} {\varphi^2\over | x | } dx ,
\int_{\mathbb{R}^3} |\varphi |^2  = 1 \Big\} < 0\,
$$
i.e.\eqref{1.7} is satisfied also.

\medskip
Our main result is the following. We prove that the
Schr\"odinger--Poisson system \eqref{1.1}-\eqref{1.2} has a ground
state, minimizing the energy functional corresponding to
\eqref{1.3},  given by (see Lemma \ref{lm2.2}):
\begin{equation}
E(\varphi) := {1\over 4}\int_{\mathbb{R}^3}| \nabla \varphi |^2 dx
+{1\over 4}\int_{\mathbb{R}^3}|\nabla V(\varphi)|^2 dx +\frac
12\int_{\mathbb{R}^3} \widetilde V \varphi^2 dx +{\omega\over
2}\int_{\mathbb{R}^3} \varphi^2 dx  \label{1.8}
\end{equation}

\begin{theorem} \label{thm1.3}
Under the assumptions \eqref{1.5}, \eqref{1.6}, and \eqref{1.7}
there exists $\omega_* > 0$ such that for all $0 <\omega <
\omega_*$ the equation \eqref{1.3} has a nonnegative solution
$u\not\equiv 0$ which minimizes the functional $E$:
$$ E(u) = \min_{ \varphi \in H^1 (\mathbb{R}^3)} E (\varphi ) .
$$
\end{theorem}

The remainder of this paper is organized as follows: In section 2
we present some preliminary lemmas which will be useful in the
sequel. In section 3, we conclude by proving our main result.

\section{Preliminary results}

In this section we present a few preliminary lemmas which shall be
required in several proofs. Recall (cf. [7, Theorem I.1] or [10,
p.151]) that ${\cal D}^{1,2} (\mathbb{R}^3)$ is the completion of
$C_0^\infty (\mathbb{R}^3)$ for the norm
$$\| \varphi \|_{{\cal D}^{1,2}} =
\Big(\int_{\mathbb{R}^3} |\nabla \varphi|^2 \, dx \Big)^{1/2} .
$$
By a Sobolev inequality, ${\cal D}^{1,2} (\mathbb{R}^3)$ is
continuously embedded in $L^6 (\mathbb{R}^3)$, an equivalent
characterization is
$$ {\cal D}^{1,2} (\mathbb{R}^3):= \left\{ \varphi \in L^6 (\mathbb{R}^3);
  |\nabla \varphi | \in L^2 (\mathbb{R}^3) \right\} .
$$
For the solvability of the Poisson equation \eqref{1.1} we state
the following lemma.

\begin{lemma} \label{lm2.1}
For all $f\in L^{6/5} (\mathbb{R}^3)$, the equation
\begin{equation}
- \Delta W  =  f \quad \mbox{in }  \mathbb{R}^3 \label{2.1}
\end{equation}
has a unique solution $W \in{\cal D}^{1,2} (\mathbb{R}^3)$ given
by
\begin{equation}
W(f)(x)= {1\over{4\pi}}\int_{\mathbb{R}^3} {f(y)\over {| x-y|}}
\,d y \,. \label{2.2}
\end{equation}
\end{lemma}

\paragraph{Proof.}  The existence and the uniqueness of the
solution of \eqref{2.1} follow from corollary 3.1.4 of reference
[5], by minimizing on ${\cal D}^{1,2} (\mathbb{R}^3)$ the
functional
$$
J (v) =  \frac 12 \int_{\mathbb{R}^3} | \nabla v |^2 dx -
\int_{\mathbb{R}^3} f  v dx .
$$
For this, using H\"older's and Sobolev's inequalities we check
easily that $J$ is coercive (that is $J (v_n) \to + \infty  $ as
$\| v_n \|_{{\cal D}^{1,2}}\to \infty $), strictly convex, lower
semi-continuous and $C^1$ on ${\cal D}^{1,2} (\mathbb{R}^3)$.
Hence $J$ attains its minimum at $W\in {\cal D}^{1,2}
(\mathbb{R}^3)$ which is the unique solution of \eqref{2.1}.


By uniqueness, $W$ is the Newtonian potential of $f$ and has (cf.
[1, p.235]) an explicit formula given by \eqref{2.2}. Furthermore,
multiplying \eqref{2.1} by $W$ and integrating we obtain
$$\|\nabla W\|^2 = \int_{\mathbb{R}^3} f(x) W(x) dx .
$$
After using H\"older and Sobolev inequalities we get
\begin{equation}
\| \nabla W \| \; \leq  \; S_*^{1/2}  \| f \|_{L^{6/5}}
\label{2.3}
\end{equation}
where $S_*$ is the best Sobolev constant in
\begin{equation}
\| v \|_{L^6 (\mathbb{R}^3)}^2  \leq
  S_* \| \nabla v \|_{L^2 (\mathbb{R}^3)}^2 .\label{2.4}
\end{equation}
Hence the linear mapping $ f\mapsto W $ is continuous from
$L^{6/5} (\mathbb{R}^3)$ into ${\cal D}^{1,2} (\mathbb{R}^3)$.
\begin{flushright}$\square$ \end{flushright}
\smallskip

Now in order to find a solution of equation \eqref{1.3}, we are
going to show that the operator
$$
v \mapsto -\frac 12 \Delta v  +  ( W(|v|^2 - n^*) + \widetilde V )
v +  \omega  v
$$
is the derivative of a functional $I: H^1 (\mathbb{R}^3) \to
\mathbb{R}$ and hence equation \eqref{1.3} has a variational
structure. To this end, we have the following lemma (see also [3])

\begin{lemma} \label{lm2.2}
Let $n^*\in L^{6/5}(\mathbb{R}^3)$. For $\varphi\in
H^1(\mathbb{R}^3)$ we denote by $V (\varphi):= W(|\varphi|^2 -
n^*)$ the unique solution of \eqref{2.1} when $ f:= |\varphi|^2 -
n^* $ . Define
$$I(\varphi) :=  {1\over 4} \int_{\mathbb{R}^3} |\nabla V (\varphi)  |^2 dx
.$$ Then $ I $ is $C^1$ on $H^1 (\mathbb{R}^3)$ and its derivative
is given by
\begin{equation}
\langle I^{'}(\varphi) , \psi \rangle  =  \int_{\mathbb{R}^3} V
(\varphi) \varphi \psi dx \quad  \forall \psi \in H^1
(\mathbb{R}^3) .\label{2.5}
\end{equation}
\end{lemma}

\paragraph{Proof.}  Note that if $\varphi\in H^1
(\mathbb{R}^3)$ then, by interpolation, $|\varphi|^2 \in L^{6/5}
(\mathbb{R}^3)$. So taking $ f = | \varphi |^2  - n^* $  and
multiplying the equation \eqref{2.1}  by  $ V (\varphi):=
W(|\varphi|^2 - n^*)$ we deduce that $\|\nabla V (\varphi)\|^2 =
\int f(x) V (\varphi) (x) dx $, and hence in view of \eqref{2.2}
we get
\begin{equation}
I (\varphi) = {1 \over {16\pi}} \int\int { (|\varphi|^2 - n^* )
(x) (| \varphi |^2 - n^* )(y) \over {|x-y|}} \,dx \,dy .
\label{2.6}
\end{equation}
Using this expression, we show easily that \eqref{2.5} holds for
the G\^ateaux differential of $I$ i.e. for all $\varphi, \; \psi
\in H^1 (\mathbb{R}^3)$
$$
\lim_{t\to 0^+} {{I (\varphi + t \psi) - I (\varphi)}\over t} =
\int_{\mathbb{R}^3} V (\varphi) \varphi \psi\, dx ,
$$
and that the mapping $\varphi \mapsto \varphi V(\varphi)$ is
continuous on $H^1 (\mathbb{R}^3)$. Thus $ I $ is Frechet
differentiable and $C^1$ on $H^1 (\mathbb{R}^3)$ and its
derivative satisfies \eqref{2.5}. \hfill$\square$

At certain steps of our proof of Theorem \ref{thm1.3}, we need
some estimates for which we will use the next inequalities.

\begin{lemma} \label{lm2.3}
$(i)$ If $ \theta \in L^r (\mathbb{R}^3)$ for some $  r \geq
{3/2}$  then $\forall \delta > 0 ,  \exists C_\delta > 0$ such
that
\begin{equation}
\int_{\mathbb{R}^3} \theta (x) |\varphi (x) |^2 dx  \leq  \delta
\| \nabla \varphi \|^2   +   C_\delta \| \varphi \|^2 \quad
\forall \varphi \in H^1 (\mathbb{R}^3) \label{2.7}
\end{equation}
$(ii)$ For all $\varphi\in {\cal D}^{1,2} (\mathbb{R}^3) $ and
$y\in \mathbb{R}^3$ one has
\begin{equation}
\int_{\mathbb{R}^3} {{|\varphi (x) |^2}\over {| x-y |^2}} dx \leq
4 \| \nabla \varphi \|^2 \label{2.8}
\end{equation}
$(iii)$ For any $ \delta > 0 $ and all $y\in \mathbb{R}^3$
\begin{equation}
\int_{\mathbb{R}^3} {{|\varphi (x) |^2}\over {| x-y |}} dx \leq
  \delta \| \nabla \varphi \|^2  +   {4\over \delta} \| \varphi \|^2
  \quad \forall \varphi \in H^1 (\mathbb{R}^3) \label{2.9}
\end{equation}
\end{lemma}

\paragraph{Proof.}  In order to prove $(i)$ we show first
that \eqref{2.7} holds for any $\theta \in L^{\infty} +  L^{3/2} $
and conclude since $L^r (\mathbb{R}^3) \subset  L^\infty
(\mathbb{R}^3) + L^{3/2}(\mathbb{R}^3) $ for all $r \geq {3/2}$.
Let $\theta = \theta_1 + \theta_2$ with $\theta_1 \in L^{\infty}$
and $\theta_2 \in L^{3/2}$. Then for each $\lambda > 0$ we have
\begin{align*}
\int_{\mathbb{R}^3} \theta (x) |\varphi (x) |^2 dx \leq  & \|
\theta_1\|_{L^\infty} \| \varphi\|^2 + \lambda \int_{[|\theta_2 |
\leq \lambda]} |\varphi |^2 dx +  \int_{[ |\theta_2 | >
\lambda]} |\theta_2 | |\varphi |^2 dx\\
\leq  & \left(\| \theta_1\|_{L^\infty}  +  \lambda \right) \|
\varphi\|^2    +   \|\theta_2 \|_{L^{3/2}([ |\theta_2 | >
\lambda ])} \|\varphi\|_{L^6}^2 \\
\leq   & \left(\| \theta_1\|_{L^\infty} + \lambda \right) \|
\varphi\|^2 +   S_* \|\theta_2^\lambda \|_{L^{3/2}}
\|\nabla\varphi\|^2
\end{align*}
where $S_*$ is the best Sobolev constant in \eqref{2.4} and
$\theta_2^\lambda$ denotes $\theta_2^\lambda:= \theta_2
\mathbb{I}_{[|\theta_2 | > \lambda]}$. It is clear that
$|\theta_2^\lambda | \leq |\theta_2 |$ for all $\lambda > 0$ and
that $\theta_2^\lambda \to 0 $ pointwise a.e. when $\lambda \to
+\infty $. Since $\theta_2 \in L^{3/2}$ then  by Lebesgue
convergence theorem we infer that $ \| \theta_2^\lambda
\|_{L^{3/2}}$ converges to zero. Hence for any $\delta > 0$ there
exists $ K_\delta > 0$ such that if $ \lambda \geq K_\delta$ one
has  $ S_* \|\theta_2^\lambda \|_{L^{3/2}}  \leq  \delta $.
Choosing $C_\delta :=\|\theta_1 \|_{L^\infty} + K_\delta $ we
deduce that \eqref{2.7} holds for all $\theta \in L^\infty
(\mathbb{R}^3) + L^{3/2}(\mathbb{R}^3)$.

\noindent Regarding $(ii)$, \eqref{2.8} is the classical Hardy
inequality (see [2]).

\noindent Finally, to show $(iii)$ for all $\delta > 0$ and any $
y \in \mathbb{R} $, we write
\begin{align*}
\int_{\mathbb{R}^3} {{|\varphi (x) |^2}\over {| x-y |}} dx = &
\int_{| x-y|<{\delta \over 4}}{{|\varphi (x) |^2}\over {| x-y
|^2}} | x-y| dx +  \int_{| x-y|\geq {\delta \over 4}}{{|\varphi
(x) |^2}\over {| x-y |}} dx \\
\leq   & {\delta \over 4} \int_{\mathbb{R}^3} {{|\varphi (x)
|^2}\over {| x-y |^2}} dx  + {4\over \delta} \int_{\mathbb{R}^3}
|\varphi (x) |^2 dx
\end{align*}
and \eqref{2.9} holds by using Hardy inequality \eqref{2.8}.
\hfill$\square$

\begin{remark} \label{rmk2.4} \rm
Note that  ${\widetilde V}^-$ satisfies the inequality \eqref{2.7}
i.e. $\forall \delta > 0   \exists C_\delta > 0$ such that
\begin{equation}
\int_{\mathbb{R}^3} {\widetilde V}^- (x) |\varphi (x) |^2 dx  \leq
\delta  \| \nabla \varphi \|^2   +   C_\delta \| \varphi \|^2
\quad \forall \varphi \in H^1 (\mathbb{R}^3) .\label{2.10}
\end{equation}
Indeed, by $(1.7)\; {\widetilde V}^-$ satisfies the decomposition
\eqref{1.4}. Then for a fixed $\lambda > 0$ we have
$${\widetilde V}^- = {\widetilde V}_{1\lambda}^- + {\widetilde
V}_{2\lambda}^-
$$
where for $i=1, 2$, ${\widetilde V}_{i\lambda}^- \in L^s
(\mathbb{R}^3)$ for some $s\in [3/2 , \infty]$ ($s=q_0$ or
$s=q_\lambda$). Hence by Lemma \ref{lm2.3} each ${\widetilde
V}_{i\lambda}^-$ satisfies the inequality \eqref{2.7} and
consequently ${\widetilde V}^-$ also.
\end{remark}

To finish this section we state the following convergence Lemma.

\begin{lemma} \label{lm2.5}
Let $\psi \in L^r(\mathbb{R}^3)$ for some $  r > {3/2}$. If $v_n
\rightharpoonup 0$ weakly in $H^1 (\mathbb{R}^3)$ then
$$\int_{\mathbb{R}^3} \psi (x)  v^2_n (x) dx \to 0 \quad  as \quad
n \to +\infty $$
\end{lemma}

\paragraph{Proof.}  Consider the subset of $\mathbb{R}^3$,
$A_\lambda:= [|\psi| > \lambda ]$ and a compact subset $K$ of
$A_\lambda$ suitably chosen later. We write
\begin{align*}
\int_{\mathbb{R}^3} |\psi|(x)  v^2_n (x) dx &=\int_{\mathbb{R}^3 -
A_\lambda} |\psi|  v^2_n  dx  +
\int_{A_\lambda - K} |\psi| v^2_n    dx   +  \int_{K} |\psi| v^2_n \,dx \\
&\leq    \lambda \| v_n \|^2 +  \| \psi\|_{L^r (A_\lambda - K)} \|
v_n \|_{L^{2r'}(\mathbb{R}^3)}^2 + \| \psi \|_{L^r (\mathbb{R}^3)}
\| v_n \|_{L^{2r'}(K)}^2 \cr\noalign {\medskip} &\leq   \lambda
C_0  +  C_1 \| \psi \|_{L^r (A_\lambda - K)} +  \| \psi \|_{L^r
(K)} \| v_n\|_{L^{2r'}(K)}^2
\end{align*}
where $ {1 \over {r'}} + {1 \over r} = 1$. In the last inequality
we used  that $ (v_n)_n $ is bounded in $H^1 (\mathbb{R}^3)$ (note
that $2<2r'<6$). For a given arbitrary $\delta > 0$, we fix first
$\lambda$ such that $\lambda C_0 \leq {\displaystyle {\delta \over
3}}$. Next we choose a compact subset $ K\subset A_\lambda$ such
that
$$ C_1  \| \psi \|_{L^r (A_\lambda - K)}  \leq  {\delta \over 3}
$$
and finally since $ v_n \rightharpoonup 0$ in $ H^1
(\mathbb{R}^3)$ and $ 2 < 2 r' < 6$ then up a subsequence  $\|
v_n\|_{L^{2r'}(K)}^2$ converges to $0$ and therefore there exists
$N_\delta \in \mathbb{N}$ such that for all $n\geq N_\delta$ we
get
$$\| \psi \|_{L^r (K)} \| v_n\|_{L^{2r'}(K)}^2  \leq {\delta \over 3}
$$
which completes the proof. \hfill$\square$

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

Now we are in position to prove our main result. To this end, we
shall minimize the energy functional
$$ E (\varphi) := {1\over 4}\int|\nabla \varphi |^2 dx +  I(\varphi) +
\frac 12 \int \widetilde V \varphi^2 dx  + {\omega\over 2} \int
\varphi^2 dx
$$
whose critical points correspond, on account of Lemma \ref{lm2.2},
to solutions of \eqref{1.3}. Using \eqref{2.6}, we may decompose $
E(\varphi)$ as
\begin{equation}
E(\varphi)= E_1(\varphi) -  E_2(\varphi) +  E_3(\varphi) +  E(0)
\label{3.1}
\end{equation}
where
\begin{align*}
E_1(\varphi):= & {1\over 4}\int|\nabla\varphi|^2 \,dx + {1\over2}
\int\widetilde V^+\varphi^2 dx + {\omega\over2}\int \varphi^2 \,dx \\
  E_2(\varphi):= & {1\over2} \int \widetilde V^- \varphi^2 \, dx
  + {1\over 8 \pi}\int\!\int{n^* (y) \over {|x-y|}}  \varphi^2 (x) \,dx\, dy
\\
E_3(\varphi):= &{1\over16\pi} \int\!\int{\varphi^2(x)
\varphi^2(y)\over{|x-y|}}
\,dx \, dy \\
E(0):= & {1\over16\pi} \int\!\int{n^*(x) n^*(y)\over{|x-y|}}
\,dx\, dy .
\end{align*}
The proof of Theorem \ref{thm1.3} is divided into the four
following Lemmas:

\begin{lemma} \label{lm3.1}
Let $\omega > 0 $ and $c \in \mathbb{R}$. If the set $ [ E \leq c
] $ is bounded in $L^2 (\mathbb{R}^3)$ then it is also bounded in
$H^1 (\mathbb{R}^3)$.
\end{lemma}

\paragraph{Proof.}  By the expression \eqref{3.1}, $ E(\varphi) \leq  c$
implies in particular
\begin{equation}
{1\over 4}\|\nabla \varphi \|^2  -  E_2 (\varphi)   \leq    c_0
\label{3.2}
\end{equation}
where $c_0 := c - E(0)$ and since the other terms are nonnegative.
To estimate $E_2 (\varphi)$ we use \eqref{2.9} which gives for any
$\delta > 0$,
$$\int\!\int_{\mathbb{R}^3\times\mathbb{R}^3} {n^* (y) \over {|x-y|}}
\varphi^2 (x) dx dy \leq  \left( \delta \| \nabla \varphi \|^2  +
   {4\over \delta} \| \varphi \|^2  \right) \| n^* \|_{L^1} .
$$
Using this inequality, Remark \ref{rmk2.4} and choosing $\delta$
such that ${\delta \big( \frac 12 + {\| n^* \|_{L^1} \over {8\pi}}
\big) < {1\over 8}}$ we obtain
\begin{equation}
E_2 (\varphi) \leq   {1\over 8} \|\nabla \varphi \|^2 +  K_0
\|\varphi \|^2 \label{3.3}
\end{equation}
where $K_0$ is a positive constant. In Consequence \eqref{3.2}
gives
$$
{1\over 8}\|\nabla \varphi \|^2 \leq  K_0  \|\varphi \|^2  + c_0 .
$$

\begin{lemma} \label{lm3.2}
For all $\omega > 0 $ and $c\in \mathbb{R} $ the set $ [ E \leq c
] $ is bounded in $L^2 (\mathbb{R}^3)$.
\end{lemma}

\paragraph{Proof.}  Assume by contradiction that there
exists a sequence $(u_j)_j \subset H^1 (\mathbb{R}^3)$ such that
$E(u_j) \leq c$ and $\| u_j \| \to +\infty$. Let $v_j := u_j/ \|
u_j \|$ then  $\| v_j \| = 1$ and from $E(u_j) \leq c$ we get
\begin{equation}
{1\over 4} \int |\nabla v_j |^2 dx  -  E_2 (v_j) +  E_3 (v_j)
\|u_j\|^2  + {\omega\over 2}  \leq    {c_0 \over \| u_j \|^2} .
\label{3.4}
\end{equation}
By using the estimate \eqref{3.3} for $\varphi:=v_j$  we obtain
\begin{equation}
{1\over 8} \|\nabla v_j \|^2    +    E_3 (v_j) \|u_j\|^2 +
{\omega\over 2} \leq    {c_0 \over \| u_j \|^2}   +  K_0 .
\label{3.5}
\end{equation}
Since $ \omega$ and $E_3 (v_j)$ are nonnegative, this inequality
implies that $(v_j)_j$ is bounded in $H^1 (\mathbb{R}^3)$ and that
$E_3 (v_j) \|u_j\|^2 $ is also bounded; i.e.
$$
\Big( \int\!\int_{\mathbb{R}^3\times \mathbb{R}^3} {v_j^2 (x)
v_j^2 (y) \over{| x - y |}} dx dy \Big) \| u_j \|^2 \leq   c_1 .
$$
Let then $v \in H^1 (\mathbb{R}^3)$ be such that for a subsequence
of $v_j$, noted again $v_j$, we have $v_j \rightharpoonup v$
weakly in  $H^1 (\mathbb{R}^3)$, $ v_j \to v $ pointwise almost
everywhere and $v_j^2$ converging  to $v^2$ strongly in  $L^p_{\rm
{loc}} (\mathbb{R}^3)$ for any $1 \leq p < 3$. By Fatou's Lemma we
deduce that
\begin{align*}
\int\!\int_{\mathbb{R}^3\times \mathbb{R}^3} {v^2 (x)  v^2 (y)
\over{| x - y |}} dx dy \leq &  \liminf_{j\to +\infty}
\int\!\int_{\mathbb{R}^3\times \mathbb{R}^3} {v_j^2 (x)
v_j^2 (y) \over{|x - y|}}\, dx\, dy \\
\leq&   \liminf_{j\to +\infty}   {c_1 \over \| u_j \|^2} = 0
\end{align*}
and therefore $ v\equiv 0 $. On the other hand, it follows from
\eqref{3.4} that
\begin{equation}
{\omega\over 2} -  E_2 (v_j) \leq  {c_0 \over \| u_j \|^2}
.\label{3.6}
\end{equation}
Set
\begin{equation}
h(x) : =  {\widetilde V}^{-} (x) + V^* (x) \label{3.7}
\end{equation}
where $ V^* (x):=  {1\over 4\pi}\int {n^* (y) \over{| x - y |}} dy
$ is the Newtonian potential of $n^*$ given by Lemma \ref{lm2.1} .
Then \eqref{3.6} is equivalent to
\begin{equation}
\omega -  \int_{\mathbb{R}^3} h(x)  v_j^2 (x) dx \leq  {2 c_0
\over \| u_j \|^2} \,.\label{3.8}
\end{equation}
Using successively the hypothesis \eqref{1.5} and Lemma
\ref{lm2.5} we may show that
\begin{equation}
{\int_{\mathbb{R}^3} h(x)  v_j^2 (x) dx} \to 0\quad\mbox{as } j
\to +\infty . \label{3.9}
\end{equation}
Passing to the limit in \eqref{3.8} we infer that $\omega \leq 0$
which is a contradiction. In conclusion, any $(u_j)_j\subset H^1
(\mathbb{R}^3)$ such that $ E (u_j) \leq  c$ is bounded in $L^2
(\mathbb{R}^3)$. \hfill$\square$

\begin{lemma} \label{lm3.3}
For any $ \omega > 0 $ the functional $ E $ is weakly lower
semi-continuous on $ H^1 (\mathbb{R}^3) $ and attains its minimum
on $ H^1 (\mathbb{R}^3)$ at $ u \geq 0$.
\end{lemma}

\paragraph{Proof.}  First, to show that the functional $E$
is weakly lower semi-continuous, remark that in the expression
\eqref{3.1} the term $E_1$ and $E_3$ are continuous and convex
(therefore weakly lower semi-continuous). Then we just have to
prove that $ u \mapsto \int_{\mathbb{R}^3} h(x)  u^2 (x) dx  $ is
weakly sequentially continuous on  $H^1 (\mathbb{R}^3)$ where $ h
$ is defined by \eqref{3.7}. Consider $ u_j \rightharpoonup u $
weakly in $ H^1 (\mathbb{R}^3)$ and write
$$\int h(x)  u_j^2 (x) dx  =  \int h(x) ( u_j - u )^2 dx
+    2 \int h(x) u ( u_j - u )  dx  +  \int h(x)  u^2  dx .
$$
Taking $( u_j - u )$ instead of $v_j$ in $(3.9)$ we infer that
$$\int_{\mathbb{R}^3} h(x) ( u_j - u )^2 dx \to 0 \quad \mbox{as }
j \to \infty .
$$
Moreover, similarly to the proof of $(3.9)$ we show that
$$ \int_{\mathbb{R}^3} h(x) u ( u_j - u )  dx \to 0 \quad \mbox{as }
  j\to \infty ,
$$
and consequently
$$
\int_{\mathbb{R}^3} h(x) u_j^2 (x) dx \to \int_{\mathbb{R}^3} h(x)
u^2 (x) dx \quad {\rm as} \quad j \to \infty .
$$
This means that $u\mapsto \int_{\mathbb{R}^3} h(x) u^2 (x) dx $ is
weakly sequentially continuous on  $H^1 (\mathbb{R}^3)$ and
therefore $ E $ is weakly lower semi-continuous on $H^1
(\mathbb{R}^3)$.

Next, if we denote $ \mu := \inf  \left\{ E (\varphi) ;  \varphi
\in H^1 (\mathbb{R}^3) \right\}$ and $ (u_n)_n \subset H^1
(\mathbb{R}^3)$ a minimizing sequence then by Lemmas \ref{lm3.1}
and \ref{lm3.2},  $ (u_n)_n$ is bounded in $ H^1 (\mathbb{R}^3)$
and therefore there exists $ u \in H^1 (\mathbb{R}^3)$ such that
$ u_n \rightharpoonup u$ weakly in $  H^1 (\mathbb{R}^3)$. The
functional $E$ being  weakly lower semi-continuous on $ H^1
(\mathbb{R}^3)$ we have
$$ E(u) \leq  \liminf_{n\to +\infty}  E (u_n) = \mu
$$
and consequently $ E(u) = \mu $. Since $E$ is $C^1$ on $H^1
(\mathbb{R}^3)$ then $ E^{'}(u) = 0$ and in view of Lemma
\ref{lm2.2},  $u$ is a solution of the equation \eqref{1.3}.

Let us remark finally that by a simple inspection we have $ E (
|u| ) \leq  E (u)$  and therefore we may assume that $ u \geq 0 $
. \hfill$\square$


\begin{lemma} \label{lm3.4}
There exists $ \omega_* > 0 $ such that if $ 0< \omega < \omega_*$
then $ E (u) < E (0) $ and thus $u\not\equiv 0$.
\end{lemma}

\paragraph{Proof.}  Assuming \eqref{1.7}, there exist $ \mu_1  <0$ and
$ \varphi_1 \in H^1 (\mathbb{R}^3)$ such that $\int|\varphi_1 |^2
= 1 $ and
$$
\int_{\mathbb{R}^3} |\nabla \varphi_1 |^2  dx   +
\int_{\mathbb{R}^3} \varrho (x) \varphi_1^2(x)  dx    <    \mu_1 .
$$
From \eqref{3.1} we observe that
$$\int_{\mathbb{R}^3} |\nabla \varphi |^2  dx  +  \int_{\mathbb{R}^3}
\varrho (x) \varphi^2(x)  dx = 4 E_1 (\varphi) - 4 E_2 (\varphi) -
2 \omega \int_{\mathbb{R}^3} \varphi^2(x)  dx .
$$
Then the last inequality gives
$$E_1 (\varphi) - E_2 (\varphi) - {\omega\over 2} \, < \, {\mu_1\over 4} .$$
Now, for $ t > 0$ and using again \eqref{3.1} we compute easily
\begin{align*}
E ( t \varphi_1)  -   E(0)   = & \, t^2 E_1 (\varphi_1)  -  t^2
E_2 (\varphi_1)
+  t^4 E_3 (\varphi_1)\\
< &  {t^2 \over 4} \left[  (\mu_1  +  2 \omega )  +  4 t^2 E_3
(\varphi_1) \right] .
\end{align*}
Hence, if $  ( \mu_1  +  2 \omega )  <   0  $  there exists $ t_*
>0$ small enough such that for all $ 0 < t \leq t_* $,
$$  (\mu_1  +  2 \omega )  +  4 t^2 E_3 (\varphi_1)<0\,.
$$
In other words, setting $ \omega_* := - {\mu_1} / 2 $ then if $ 0
< \omega < \omega_*$ we have $  E (t \varphi_1)  < E(0) $  for $ 0
< t \leq t_* $.  Since $ E (u) := \inf \{ E (\varphi) ;  \varphi
\in H^1 (\mathbb{R}^3) \}$, this implies that $ E (u) <  E (0) $
and consequently $ u\not\equiv 0$. The proof of Theorem
\ref{thm1.3} is thus complete. \hfill$\square$

\begin{remark} \label{rmk3.5} \rm
If $ n^*$ is nonnegative then we may replace the assumption
\eqref{1.7} by the next one
$$\inf \left\{ \int |\nabla \varphi |^2 dx  +  2 \int \widetilde V (x)
\varphi^2  dx    ;    \int |\varphi |^2  = 1  \right\}    <    0
$$
which does not depend on $n^*$ and implies obviously \eqref{1.7}.
\end{remark}

\paragraph{Acknowledgments.}
The author acknowledges the hospitality of Laboratoire de
Math\'ematiques Appliqu\'ees de Versailles (France) where a part
of this work was done. He is grateful to Otared Kavian for very
valuable discussions and suggestions.

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\noindent\textsc{Khalid Benmlih} \\
Department of Economic Sciences,
University of Fez \\
P.O. Box 42A, Fez, Morocco.\\
E-mail: kbenmlih@hotmail.com

\end{document}