Multiple positive bound state solutions for fractional Schrödinger-Poisson system with critical growth

Abstract

In this paper, we deal with the following critical fractional Schrödinger-Poisson system without subcritical perturbation $$ \begin{cases} (-\Delta)^{s} u + V(x)u + K(x)\phi u=|u|^{2_{s}^{*}-2}u, & x\in\mathbb{R}^{3},\\ (-\Delta)^{t}\phi=K(x)u^{2}, & x\in\mathbb{R}^{3}, \end{cases} $$ where $s\in(\frac{3}{4},1),$ $t\in(0,1)$, $2^{\ast}_{s}=\frac{6}{3-2s}$ is the critical Sobolev exponent. When $V(x)$ is positive bounded from below, combining with variation methods and Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this system. The results obtained in this paper extend and improve some recent works in which potential function $V(x)$ may vanish at the infinity.