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

Abstract

We study the following Schrödinger-Poisson system (P_λ){ll} -Δu + V(x)u+λϕ(x) u =Q(x)u^{p}, x\in \mathbb{R}^3 \\ -Δϕ= u^2, \lim\limits_{|x|\to +\infty}ϕ(x)=0, u>0, where $\lambda ⩾0$ 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 $V(x)\equiv Q(x)\equiv 1$, 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 $\lambda ⩾1/4$. 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 $({\mathrm{P}}_{\lambda })$ converges, along a subsequence, to a solution of ($P_0$) in $H^1$ as $\lambda \to 0$.

13 pages. to appear in Acta Math. Sci., 29B(2009), No.4, pp:1095-