Multiplicity and concentration of nontrivial solutions for fractional Schrödinger–Poisson system involving critical growth

In this paper, we study the concentration and multiplicity of solutions to following fractional Schr\"{o}dinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)+u^{2_s^{\ast}-1} & \hbox{in $\mathbb{R}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2, u>0& \end{array} \right. \end{equation*} where $s>\frac{3}{4}$, $s,t\in(0,1)$, $\varepsilon>0$ is a small parameter, $f\in C^1(\mathbb{R}^{+},\mathbb{R})$ subcritical, $V:\mathbb{R}^3\rightarrow\mathbb{R}$ continuous bounded function. We establish family positive $u_{\varepsilon}\in H_{\varepsilon}$ which concentrates around local minima $V$ in $\Lambda$ as $\varepsilon\rightarrow0$. With Ljusternik-Schnirelmann theory, also obtain multiple by employing topology construct set potential attains its minimum.