# Concentrating bounded states for fractional Schr\"{o}dinger-Poisson system involving critical Sobolev exponent

## TL;DR

This paper investigates the existence and multiplicity of positive solutions to a fractional Schr"odinger-Poisson system with critical Sobolev exponent, showing solutions concentrate near potential minima as the parameter approaches zero.

## Contribution

It establishes the concentration of solutions around local minima of the potential and employs topological methods to prove multiple solutions exist.

## Key findings

- Solutions concentrate near local minima of V as epsilon approaches zero.
- Multiple solutions are obtained using Ljusternik-Schnirelmann theory.
- The results extend understanding of fractional Schr"odinger-Poisson systems with critical exponents.

## Abstract

In this paper, we study the concentration and multiplicity of solutions to the following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)+u^{2_s^{\ast}-1} & \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 $s>\frac{3}{4}$, $s,t\in(0,1)$, $\varepsilon>0$ is a small parameter, $f\in C^1(\mathbb{R}^{+},\mathbb{R})$ is subcritical, $V:\mathbb{R}^3\rightarrow\mathbb{R}$ is a continuous bounded function. We establish a family of positive solutions $u_{\varepsilon}\in H_{\varepsilon}$ which concentrates around the local minima of $V$ in $\Lambda$ as $\varepsilon\rightarrow0$. With Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topology construct of the set where the potential $V$ attains its minimum.