Bound state solutions for non-autonomous fractional Schrödinger-Poisson equations with critical exponent

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.