\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \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. \begin{thebibliography}{00} \frenchspacing \bibitem{g} D. Gilbard \& N.S. Trudinger: {\it Elliptic Partial Differential Equations of Second Order}; 2nd edition, Springer, Berlin 1983. \bibitem{h1} G.H. 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