Bound states for a stationary nonlinear Schrodinger-Poisson system with sign-changing potential in $R^3$

Abstract

We study the following Schr\"odinger-Poisson system (P_\lambda){ll} -\Delta u + V(x)u+\lambda \phi (x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -\Delta\phi = u^2, \lim\limits_{|x|\to +\infty}\phi(x)=0, u>0, where $\lambda\geqslant0$ is a parameter, $1 < p < +\infty$, $V(x)$ and $Q(x)$ are sign-changing or non-positive functions in $ L^{\infty}(\mathbb{R}^3)$. When $V(x)\equiv Q(x)\equiv1$, D.Ruiz \cite{RuizD-JFA} proved that ($P_\lambda$) with $p\in(2,5)$ has always a positive radial solution, but ($P_\lambda$) with $p\in(1,2]$ has solution only if $\lambda>0$ small enough and no any nontrivial solution if $\lambda\geqslant{1/4}$. By using sub-supersolution method, we prove that there exists $\lambda_0>0$ such that ($P_\lambda$) with $p\in(1,+\infty)$ has always a bound state ($H^1(\mathbb{R}^3)$ solution) for $\lambda\in[0,\lambda_0)$ and certain functions $V(x)$ and $Q(x)$ in $ L^{\infty}(\mathbb{R}^3)$. Moreover, for every $\lambda\in[0,\lambda_0)$, the solutions $u_\lambda$ of $\rm (P_\lambda)$ converges, along a subsequence, to a solution of ($P_0$) in $H^1$ as $\lambda \to 0$.