Existence and concentration of positive ground state solutions for nonlinear fractional Schr\"odinger-Poisson system with critical growth

In this paper, we study the following fractional Schr"{o}dinger-Poisson system involving competing potential functions \begin{equation*} \left{ \begin{array}{ll} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=K(x)f(u)+Q(x)|u|^{2_s^{\ast}-2}u, & \hbox{in ${\mathbb{R}}^3$,} \varepsilon^{2t}(-\Delta)^t\phi=u^2,& \hbox{in ${\mathbb{R}}^3$,} \end{array} \right. \end{equation*} where $\epsilon >0$ is a small parameter, $f$ is a function of $C^1$ class, superlinear and subcritical nonlinearity, $2_s^{\ast }=\frac{6}{3-2s}$, $s>\frac{3}{4}$, $t\in (0,1)$, $V(x)$ $K(x)$ and $Q(x)$ are positive continuous function. Under some suitable assumptions on $V$, $K$ and $Q$, we prove that there is a family of positive ground state solutions with polynomial growth for sufficiently small $\epsilon >0$, of which it is concentrating on the set of minimal points of $V(x)$ and the sets of maximal points of $K(x)$ and $Q(x)$. The methods are based on the Nehari manifold, arguments of Brezis-Nirenberg and concentration compactness of P. L. Lions.