Bound states for fractional Schrödinger-Poisson system with critical exponent

This paper deals with the fractional Schrödinger-Poisson system

\begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta )^su+V(x)u+K(x)\phi u = |u|^{2_{s}^{*}-2}u, & \text{in}\ {\Bbb R}^3,\\ (-\Delta)^{t}\phi = K(x)u^2, & \text{in}\ {\Bbb R}^3, \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ s\in (\frac{3}{4}, 1) $\end{document}, \begin{document}$ t\in(0, 1) $\end{document}, \begin{document}$ \varepsilon $\end{document} is a positive parameter, \begin{document}$ 2_{s}^{*} = \frac{6}{3-2s} $\end{document} is the critical Sobolev exponent. \begin{document}$ K(x)\in L^{\frac{6}{2t+4s-3}}({\Bbb R}^3) $\end{document}, \begin{document}$ V(x)\in L^{\frac{3}{2s}}({\Bbb R}^3) $\end{document} and \begin{document}$ V(x) $\end{document} is assumed to be zero in some region of \begin{document}$ {\Bbb R}^3 $\end{document}, which means that the problem is of the critical frequency case. In virtue of a global compactness result in fractional Sobolev space and Lusternik-Schnirelman theory of critical points, we succeed in proving the multiplicity of bound states.