# American Institute of Mathematical Sciences

November  2007, 18(4): 809-816. doi: 10.3934/dcds.2007.18.809

## Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$

 1 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China, China

Received  July 2006 Revised  March 2007 Published  May 2007

In this paper, we are concerned with the following problem

(P) $-\Delta u + V(x)u+\lambda \phi (x) u =f(x,u), x\in \mathbb{R}^3$
$-\Delta\phi = u^2, \lim_{|x|\rightarrow +\infty}\phi(x)=0,$

where $\lambda >0$ is a parameter, the potential $V(x)$ may not be radially symmetric, and $f(x,s)$ is asymptotically linear with respect to $s$ at infinity. Under some simple assumptions on $V$ and $f$, we prove that the problem (P) has a positive solution for $\lambda$ small and has no any nontrivial solution for $\lambda$ large.

Citation: Zhengping Wang, Huan-Song Zhou. Positive solution for a nonlinear stationary Schrödinger-Poisson system in $R^3$. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 809-816. doi: 10.3934/dcds.2007.18.809
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