\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 26, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/26\hfil Schr\"odinger--Poisson system] {A note on a 3-dimensional stationary Schr\"odinger--Poisson system} \author[Khalid Benmlih\hfil EJDE-2004/26\hfilneg] {Khalid Benmlih} \address{Khalid Benmlih \hfill\break\indent Department of Economic Sciences, University of Fez \hfill\break\indent P.O. Box 42A, Fez, Morocco} \email{kbenmlih@hotmail.com} \date{} \thanks{Submitted July 29, 2003. Published February 24, 2004.} \subjclass[2000]{35J50, 35Q40} \keywords{Schr\"odinger equation, Poisson equation, standing wave, \hfill\break\indent variational method} \begin{abstract} In a previous paper we have proved existence of a ground state for a stationary Schr\"odinger--Poisson system in the whole space $\mathbb{R}^3$ under appropriate assumptions on the data, namely the dopant-density $n^*$ and the effective potential $\widetilde V$. In this note we show that the same result remains true under less restrictive hypotheses. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} We are concerned with existence of standing waves (i.e. solutions of the form $u(t,x)= e^{i\omega t} u(x)$ with a real constant $\omega$) for a time-dependent Schr\"odinger equation where the electric potential $V$ satisfies a linear Poisson equation. This leads to solving the stationary Schr\"odinger--Poisson 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} where the dopant-density $n^*$ and the effective potential $\widetilde V $ are given reals functions. An existence result of a solution for \eqref{1.1}--\eqref{1.2} has been established by Lions \cite{lions} in the particular case where $\widetilde V (x)= -2 /|x|$ and $n^* \equiv 0$, by Nier \cite{nier} under some assumptions on the data essentially when $ \|\widetilde V \|_{L^2}$ and $ \|n^* \|_{L^2}$ are small enough and also recently by the author \cite{benmlih} under appropriate assumptions on $\widetilde V $ and $ n^* $. In this note, we show existence of a ground state of \eqref{1.1}--\eqref{1.2} as in \cite{benmlih} but under less restrictive assumptions. More precisely, an adequate modification on the proof of the main result in [1, theorem 1.3] allows us to avoid the condition where $n^* \in L^1 (\mathbb{R}^3)$. Let us recall firstly the principal theorem and the several steps of its proof given in \cite{benmlih}: after solving explicitly the Poisson equation for any fixed $u\in H^1 (\mathbb{R}^3)$, we substitute the unique solution then obtained $V = V(u)$ in the Schr\"odinger equation \eqref{1.1} and show existence of a ground state 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} To this end, we show that the energy functional corresponding to \eqref{1.3} is exactly the expression \begin{equation}\label{functional} E(\varphi) := \frac 1{4}\int_{\mathbb{R}^3}| \nabla \varphi |^2 dx+\frac 1{4}\int_{\mathbb{R}^3}|\nabla V(\varphi)|^2 dx+\frac 1{ 2}\int_{\mathbb{R}^3} \widetilde V \varphi^2 dx+\frac\omega{2}\int_{\mathbb{R}^3} \varphi^2 dx \end{equation} and a solution of \eqref{1.3} is obtained as a minimizer of $E$ on $H^1(\mathbb{R}^3)$. Before giving the assumptions imposed on $\widetilde V $ and $ n^* $ to solve the system \eqref{1.1}-\eqref{1.2}, we recall the following concepts. \begin{definition} \label{def1.1} \rm We say that $g$ satisfies the decomposition \eqref{1.5} if: \begin{itemize} \item[(i)] $g \in L^1_{\rm loc} (\mathbb{R}^3)$, \item[(ii)] $g \geq 0$, and \item[(iii)] There exists $q_0 \in [3/2 , \infty ]$ such that for all $\lambda > 0$ there 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{1.5} \end{equation} \end{itemize} \end{definition} As interesting examples of this definition we may consider $g(x)= 1/|x|^\alpha$ for some $ 0< \alpha < 2 $ or $g\in L^r(\mathbb{R}^3)$ for some $ r > 3/2$ (taking $|g|$ if $g$ is negative). In what follows we will denote by $\|\cdot \|$ the norm $\|\cdot \|_{L^2}$ on $L^2 (\mathbb{R}^3)$ and by $ [ E \leq c ]$ the set $\{ \varphi ; E(\varphi) \leq c\}$. Consider now the following hypotheses: \begin{gather} {\widetilde V}^+ \in L^1_{\rm loc} (\mathbb{R}^3)\quad \mbox{and} \quad {\widetilde V}^- \mbox {satisfies the decomposition } \eqref{1.5} \label{H1} \\ n^* \in L^1 \cap L^{6/5} (\mathbb{R}^3) \label{H2} \\ \inf \Big\{ \int_{\mathbb{R}^3} \left(|\nabla \varphi |^2 + \varrho (x) \varphi^2 \right)dx , \int_{\mathbb{R}^3} |\varphi |^2 = 1 \Big\} < 0 \label{H3} \end{gather} where $\displaystyle\varrho (x):= 2 \widetilde V (x) - { \frac 1{2\pi} \int_{\mathbb{R}^3} \frac {n^*(y)}{|x-y|} dy } $. The main result in \cite{benmlih} is as follows. \begin{theorem} \label{thm1.2} Assuming \eqref {H1}, \eqref{H2} and \eqref{H3} 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 $ given by \eqref{functional}: $$ E(u) = \min_{ \varphi \in H^1 (\mathbb{R}^3)} E (\varphi ) .$$ \end{theorem} The proof of this theorem is divided into the four following Lemmas. \begin{lemma} \label{lem1.3} Let $\omega \geq 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} \begin{lemma} \label{lem1.4} For all $\omega > 0 $ and $c \in \mathbb{R} $ the set $ [ E \leq c ] $ is bounded in $L^2 (\mathbb{R}^3)$. \end{lemma} \begin{lemma} \label{lem1.5} For any $ \omega > 0 $ the functional $ E $ is weakly lower semicontinuous on $ H^1 (\mathbb{R}^3) $ and attains its minimum on $ H^1 (\mathbb{R}^3)$ at $ u \geq 0$. \end{lemma} \begin{lemma} \label{lem1.6} There exists $ \omega_* > 0 $ such that if $ 0< \omega < \omega_*$ then $ E (u) < E (0) $ and thus $u\not\equiv 0$. \end{lemma} After analyzing the proofs of the four Lemmas above given in \cite{benmlih}, we remark that theorem \ref{thm1.2} remains true even if we replace the condition \eqref{H2} by \begin{equation} n^* \in L^{6/5} (\mathbb{R}^3) .\label{H2'} \end{equation} In the sequel we shall minimize the energy functional $E$ on the space $$ H := \big\{ u\in H^1 (\mathbb{R}^3) : \int_{\mathbb{R}^3} {\widetilde V}^+ u^2 dx < \infty \big\} $$ which is a Hilbert space, continuously embedded in $H^1 (\mathbb{R}^3)$, when endowed it with its natural scalar product and norm $$ (\varphi|\psi ):= \int_{\mathbb{R}^3}\left( \nabla\varphi \cdot \nabla\psi + \varphi \psi + {\widetilde V}^+ \varphi \psi \right) dx, \quad \|\varphi\|_H := (\varphi|\varphi)^{1/2} . $$ Consequently Theorem \ref{thm1.2} becomes \begin{theorem} \label{thm1.7} Assuming \eqref {H1}, \eqref{H3} and \eqref{H2'} 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 on the space $H $ the functional $ E $: $$ E(u) = \min_{ \varphi \in H } E (\varphi ) .$$ \end{theorem} \section {Preliminaries} Here we recall the three following Lemmas which will be useful in the sequel. \begin{lemma} \label{lem2.1} Let $n^*\in L^{6/5}(\mathbb{R}^3)$. For all $\varphi\in H^1(\mathbb{R}^3)$ the Poisson equation \begin{equation}\label{2.1} - \Delta V = |\varphi|^2 - n^* \quad \mbox{in } \mathbb{R}^3 \end{equation} has a unique solution $V:=V(\varphi) \in\mathcal{D}^{1,2} (\mathbb{R}^3)$ given by \begin{equation}\label{2.2} V(\varphi) (x)= \frac 1{{4\pi}}\int_{\mathbb{R}^3} \frac {(|\varphi|^2 - n^* )(y)}{ | x-y |} dy. \end{equation} Moreover if we denote by $$I(\varphi) := \frac 1{4} \int_{\mathbb{R}^3} |\nabla V (\varphi) |^2 dx,$$ then $I$ is $C^1$ on $H^1 (\mathbb{R}^3)$ and its derivative satisfies $$ \langle I' (\varphi) , \psi \rangle = \int_{\mathbb{R}^3} V (\varphi) \varphi \psi dx \quad \forall \psi \in H^1 (\mathbb{R}^3). $$ \end{lemma} For the proof of this lemma see [1, Lemma 2.1, Lemma 2.2]. This Lemma shows in particular that the energy functional corresponding to \eqref{1.3} is exactly the expression given in \eqref{functional}, namely $$ E(\varphi) := \frac 1{ 4}\int_{\mathbb{R}^3}| \nabla \varphi |^2 dx+I(\varphi)+\frac 1{ 2}\int_{\mathbb{R}^3} \widetilde V \varphi^2 dx+ \frac \omega {2}\int_{\mathbb{R}^3} \varphi^2 dx . $$ \begin{lemma} \label{lem2.2} Let $\theta \in L^r (\mathbb{R}^3)$ for some $ r \geq {3/2}$ then for all $\delta > 0$ there exists $C_\delta > 0$ such that $$ \int_{\mathbb{R}^3} \theta (x) |\varphi (x) |^2 dx \leq \delta \|\nabla \varphi \|^2 + C_\delta \| \varphi \|^2 \quad \forall \varphi\in H^1 (\mathbb{R}^3) . $$ \end{lemma} For the proof of this lemma see \cite{benmlih} or \cite{brezis}. Remark that since ${\widetilde V}^-$ satisfies the decomposition \eqref{1.5} then for any 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 taking $\theta:= {\widetilde V}_{i\lambda}^-$ the inequality of Lemma \ref{lem2.2} holds for $i=1,2$ and consequently for all $\delta > 0$ there exists $C_\delta >0 $ so 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 {2.3} \end{equation} \begin{lemma} \label{lem2.3} 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} For the proof of this lemma see [1, Lemma 2.5]. \section {Proof of Theorem \ref{thm1.7}} We will use once again the same steps as in \cite{benmlih}. Remark at first that the proofs given in \cite{benmlih} for Lemma \ref{lem1.5} and Lemma \ref{lem1.6} do not require the hypothesis $ n^* \in L^1 (\mathbb{R}^3)$ and consequently remain valid assuming \eqref{H2'} instead of \eqref{H2}. \begin{proof}[Proof of Lemma \ref{lem1.3}] We show here that if the set $$ [ E \leq c ] := \{ \varphi \in H ; E(\varphi) \leq c\}$$ is bounded in $L^2(\mathbb{R}^3)$ then it is also bounded in $H$. Indeed since $I(\varphi)$ and $\omega$ are both nonnegative, the inequality $E(\varphi)\leq c$ gives in particular $$ \frac 1{ 4}\|\nabla \varphi \|^2 + \frac 1{2}\int\widetilde V^+\varphi^2 dx - \frac 1{2}\int\widetilde V^-\varphi^2 dx \leq c . $$ Now using the estimate \eqref{2.3} with $\delta =1/4 $ we get $$ \frac 1{ 8}\|\nabla \varphi \|^2 + \frac 1{2}\int\widetilde V^+\varphi^2 dx \leq K_0 \|\varphi \|^2 + c . $$ for some constant $K_0 > 0$. \end{proof} Let us recall that in \cite{benmlih} we have decomposed the expression of $E(\varphi)$ as $$ E(\varphi)= E_1(\varphi) - E_2(\varphi) + E_3(\varphi) + E(0) $$ where \begin{gather*} E_1(\varphi):= \frac 1{ 4}\int|\nabla\varphi|^2 \,dx + \frac 1{2} \int\widetilde V^+\varphi^2 dx + \frac{\omega}{2}\int \varphi^2 \,dx \\ E_2(\varphi):= \frac 1{2} \int \widetilde V^- \varphi^2 \, dx + \frac 1{ 8 \pi}\iint\frac{n^* (y)}{|x-y|} \varphi^2 (x) \,dx\, dy\\ E_3(\varphi):= \frac 1{16\pi} \iint\frac{\varphi^2(x)\varphi^2(y)}{|x-y|}\,dx \, dy \\ E(0):= \frac 1{16\pi} \iint\frac{n^*(x) n^*(y)}{|x-y|}\,dx\, dy . \end{gather*} Indeed, for the term $I(\varphi)$ it suffices to multiply the equation \eqref{2.1} by $V(\varphi)$, integrate by parts and use the formula \eqref{2.2}. In the proof of the similar lemma [1, Lemma 3.1] we have estimated $E_2(\varphi)$ instead of $\int\widetilde V^-\varphi^2 dx$. More precisely we have estimated the second term of $E_2(\varphi)$ by using a certain inequality of type {\sl Hardy} and the fact that $n^* \in L^1 (\mathbb{R}^3)$. We point out finally that the decomposition of $E(\varphi)$ as above remains useful for the rest of proofs. \begin{proof}[Proof of Lemma \ref{lem1.4}] Assume by contradiction that there exists a sequence $(u_j)_j \subset H$ such that $E(u_j) \leq c$ and $\| u_j \| \longrightarrow +\infty$ as $j\rightarrow +\infty$. Let $v_j := u_j/\| u_j \|$ then $\| v_j \| = 1$ and from $E(u_j) \leq c$ we get \begin{equation} \frac 1{ 4} \int |\nabla v_j |^2 dx - E_2 (v_j) + E_3 (v_j) \|u_j\|^2 + \frac\omega{2} \leq \frac {c_0} {\| u_j \|^2} \label{3.1} \end{equation} where $c_0 = c - E(0)$. To estimate $E_2 (v_j)$ it suffices to use \eqref{2.3} for the first term $\int\widetilde V^- v_j^2 dx$ . As to the second, unlike the proof in \cite{benmlih} we do not require here the assumption $n^* \in L^1(\mathbb{R}^3)$. Indeed, setting \begin{equation} V^*(x):= \iint_{\mathbb{R}^3\times \mathbb{R}^3} \frac {n^* (y)} {|x-y|} dy = - V(0)(x) \label{3.2} \end{equation} as denoted in Lemma \ref{lem2.1} we may write $$ \iint_{\mathbb{R}^3\times \mathbb{R}^3} \frac {n^* (y)}{|x-y|} v_j^2 (x) dx dy = \int_{\mathbb{R}^3} V^*(x) v_j^2 (x) dx . $$ Knowing that $V(0) \in L^6 (\mathbb{R}^3)$ we can use once more Lemma \ref{lem2.2} with $\theta:=V^*$. On the whole, we obtain in particular $$ E_2 (v_j) \leq \frac 1{ 8} \|\nabla v_j \|^2 + K_0 $$ for some positive constant $K_0$ and consequently we infer from the inequality \eqref{3.1} that $$ \frac 1{8} \|\nabla v_j \|^2 + E_3 (v_j) \|u_j\|^2 + \frac \omega{2} \leq \frac {c_0} {\| u_j \|^2} + K_0 . $$ For the remainder of the proof, we conclude exactly as in of [1, Lemma 3.2]. Precisely we show first that, up to a subsequence, $v_j \rightharpoonup 0$ weakly in $H^1 (\mathbb{R}^3)$. Next, from \eqref{3.1} it follows in particular that \begin{equation} \frac{\omega}{2} - E_2 (v_j) \leq \frac{c_0}{\| u_j \|^2}.\label{3.3} \end{equation} Using the decomposition ${\widetilde V}^- = {\widetilde V}_{1\lambda}^- + {\widetilde V}_{2\lambda}^- $ and \eqref{3.2}, we show according to Lemma \ref{lem2.3} that $E_2 (v_j)\longrightarrow 0$ as $j\rightarrow \infty$. Finally, letting $j$ go to infinity in \eqref{3.3} we obtain a contradiction since $\omega$ is positive. Consequently, all $(u_j)_j\subset H$ such that $ E(u_j) \leq c$ is bounded in $L^2 (\mathbb{R}^3)$. \end{proof} \begin{thebibliography}{00} \bibitem{benmlih} Kh. Benmlih, Stationary Solutions for a Schr\"odinger--Poisson System in $\mathbb{R}^3$; Electron. J. Diff. Eqns., Conf. 09 (2002), pp. 65-76. (http://ejde.math.swt.edu) \bibitem{brezis} H. Brezis \& T. Kato: Remarks on the Schr\"odinger operator with singular complex potentials; J. Math. Pures \& Appl. N 2, 58 (1979), 137-151. \bibitem{lions} P. L. Lions: Some remarks on Hartree equation; Nonlinear Anal., Theory Methods Appl. 5 (1981), 1245-1256. \bibitem{nier} F. Nier: Schr\"odinger--Poisson systems in dimension $d\leq 3$, the whole space case; Proceedings of the Royal Society of Edinburgh, 123A (1993), 1179-1201. \end{thebibliography} \end{document}