Existence and concentration of solution for Schr\"odinger-Poisson system with local potential

TL;DR

This paper proves the existence and concentration of ground state solutions for a nonlinear Schr"odinger-Poisson system with local potential in 3, focusing on the behavior as the parameter  approaches zero.

Contribution

It establishes new results on the existence and concentration of solutions for Schr"odinger-Poisson systems with local potentials under specific conditions.

Findings

Existence of ground state solutions for small 

Concentration behavior of solutions as  approaches zero

Summary of open problems in the field

Abstract

In this paper, we study the following nonlinear Schr\"odinger-Poisson type equation \begin{equation*} \begin{cases} -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=f(u)&\text{in}\ \mathbb{R}^3,\\ -\varepsilon^2\Delta \phi=K(x)u^2&\text{in}\ \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $V: \mathbb{R}^3\rightarrow \mathbb{R}$ is a continuous potential and $K: \mathbb{R}^3\rightarrow \mathbb{R}$ 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 $\varepsilon>0$ small. Moreover, we summarize some open problems for the Schr\"odinger-Poisson system.