Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods

In this paper, we study the following fractional Schrödinger-Poiss-on system\begin{document}$ \begin{equation*} \begin{cases}\varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u = g(u) & \hbox{in $\mathbb{R}^3$,} \\ \varepsilon^{2t}(-\Delta)^t\phi = u^2,\,\, u>0& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $\end{document}where \begin{document}$ s,t\in(0,1) $\end{document} , \begin{document}$ \varepsilon>0 $\end{document} is a small parameter. Under some local assumptions on \begin{document}$ V(x) $\end{document} and suitable assumptions on the nonlinearity \begin{document}$ g $\end{document} , we construct a family of positive solutions \begin{document}$ u_{\varepsilon}\in H_{\varepsilon} $\end{document} which concentrate around the global minima of \begin{document}$ V(x) $\end{document} as \begin{document}$ \varepsilon\rightarrow0 $\end{document} .