Concentration of bound states for fractional Schr\"{o}dinger-Poisson system via penalization methods

TL;DR

This paper investigates the existence and concentration behavior of positive solutions to a fractional Schr"{o}dinger-Poisson system, showing solutions localize near potential minima as the parameter approaches zero.

Contribution

It introduces a penalization method to construct solutions that concentrate around the minima of the potential for a fractional Schr"{o}dinger-Poisson system.

Findings

Solutions concentrate near the global minima of V(x) as epsilon approaches zero.

A family of positive solutions is constructed under certain conditions.

The method applies to systems with fractional Laplacians of order s and t.

Abstract

In this paper, we study the following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=g(u) & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,\,\, u>0& \hbox{in $\mathbb{R}^3$,} \end{array} \right. \end{equation*} where $s,t\in(0,1)$, $\varepsilon>0$ is a small parameter. Under some local assumptions on $V(x)$ and suitable assumptions on the nonlinearity $g$, we construct a family of positive solutions $u_{\varepsilon}\in H_{\varepsilon}$ which concentrates around the global minima of $V(x)$ as $\varepsilon\rightarrow0$.