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

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 $s\in (\frac{3}{4},1]$, $2_s^{\ast }=\frac{6}{3-2s}$ 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 $\mathrm{\parallel }V{\mathrm{\parallel }}_{\frac{3}{2s}}+\mathrm{\parallel }K{\mathrm{\parallel }}_{\frac{6}{6s-3}}$ is sufficiently small, we prove that the equation has at least one bound state solution.