The existence and multiplicity of solutions of a fractional Schrödinger-Poisson system with critical growth

In this paper, we study the existence and multiplicity of solutions for the following fractional Schrödinger-Poisson system: (0.1){ε2s(−Δ)su+V(x)u+ϕu=|u|2s*−2u+f(u)inℝ3,ε2s(−Δ)sϕ=u2inℝ3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{gathered} {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}u + V\left( x \right)u + \phi u = {\left| u \right|^{2_s^* - 2}}u + f\left( u \right)in{\mathbb{R}^3}, \hfill \\ {\varepsilon ^{2s}}{\left( { - \Delta } \right)^s}\phi = {u^2}in{\mathbb{R}^3}, \hfill \\ \end{gathered} \right.$$\end{document} where 34<s<1,2s∗:=63−2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{4} < s < 1, 2_s^*:=\frac{6}{3-2s}$$\end{document} the fractional critical exponent for 3-dimension, the potential V: ℝ3 → ℝ is continuous and has global minima, and f is continuous and supercubic but subcritical at infinity. We prove the existence and multiplicity of solutions for System (0.1) via variational methods.