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

<p style='text-indent:20px;'>In this paper, we study the following fractional Schrödinger-Poiss-on system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u = g(u) &amp; \hbox{in $\mathbb{R}^3$,} \\ \varepsilon^{2t}(-\Delta)^t\phi = u^2,\,\, u&gt;0&amp; \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s,t\in(0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula> is a small parameter. Under some local assumptions on <inline-formula><tex-math id="M3">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> and suitable assumptions on the nonlinearity <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula>, we construct a family of positive solutions <inline-formula><tex-math id="M5">\begin{document}$ u_{\varepsilon}\in H_{\varepsilon} $\end{document}</tex-math></inline-formula> which concentrate around the global minima of <inline-formula><tex-math id="M6">\begin{document}$ V(x) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M7">\begin{document}$ \varepsilon\rightarrow0 $\end{document}</tex-math></inline-formula>.</p>