Existence and concentration of solution for Schrödinger-Poisson system with local potential

Abstract

In this paper, we study the following nonlinear Schrödinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon^2Δu+V(x)u+K(x)ϕu=f(u)&\text{in}\ \mathbb{R}^3,\\ -\varepsilon^2Δϕ=K(x)u^2&\text{in}\ \mathbb{R}^3, \end{cases} \end{equation*} where $\epsilon >0$ is a small parameter, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a continuous potential and $K:{\mathbb{R}}^3\to \mathbb{R}$ is used to describe the electron charge. Under suitable assumptions on $V(x),K(x)$ and $f$, we prove existence and concentration properties of ground state solutions for $\epsilon >0$ small. Moreover, we summarize some open problems for the Schrödinger-Poisson system.

20 pages; minor corrections. To appear in Partial Differential Equations and Applications