# Bound state solutions for non-autonomous fractional Schr\"{o}dinger-Poisson equations with critical exponent

## TL;DR

This paper establishes the existence of bound state solutions for a non-autonomous fractional Schrödinger-Poisson equation with critical exponent, under smallness conditions on potential functions, extending understanding of fractional quantum systems.

## Contribution

It proves the existence of bound state solutions for the fractional Schrödinger-Poisson system with critical exponent, considering non-autonomous potentials and smallness conditions.

## Key findings

- Existence of at least one bound state solution under small potential norms.
- Extension of fractional Schrödinger-Poisson analysis to critical exponent case.
- Conditions on potential functions for solution existence.

## Abstract

In this paper, we study the fractional Schr\"{o}dinger-Poisson equation \begin{equation*} \ \left\{\begin{aligned} &(-\Delta)^{s}u+V(x)u+K(x)\phi u=|u|^{2^{\ast}_{s}-2}u, &\mbox{in} \ \mathbb{R}^{3},\\ &(-\Delta)^{s}\phi=K(x)u^{2},&\mbox{in} \ \mathbb{R}^{3}, \end{aligned}\right. \end{equation*} where $s\in (\frac{3}{4},1]$, $2^{\ast}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $K\in L^{\frac{6}{6s-3}}(\mathbb{R}^{3})$ and $V\in L^{\frac{3}{2s}}(\mathbb{R}^{3})$ are nonnegative functions. If $\|V\|_{\frac{3}{2s}}+\|K\|_{\frac{6}{6s-3}}$ is sufficiently small, we prove that the equation has at least one bound state solution.