# Concentrating phenomenon for fractional nonlinear Schr\"{o}dinger-Poisson system with critical nonlinearity

## TL;DR

This paper investigates the concentration behavior of positive solutions to a fractional Schrödinger-Poisson system with critical nonlinearity, showing solutions localize near potential minima as the parameter approaches zero.

## Contribution

It constructs a family of solutions that concentrate around the global minima of the potential, extending understanding of fractional systems with critical nonlinearities.

## Key findings

- Solutions concentrate near potential minima as epsilon approaches zero.
- Constructed positive solutions in fractional Sobolev space.
- Analyzed effects of critical nonlinearity on solution behavior.

## 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 suitable assumptions on potential function $V(x)$ and critical nonlinearity term $g(u)$, we construct a family of positive solutions $u_{\varepsilon}\in H^s(\mathbb{R}^3)$ which concentrates around the global minima of $V$ as $\varepsilon\rightarrow0$.