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

Abstract

In this paper, we study the following fractional Schrödinger-Poisson system \begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}(-Δ)^su+V(x)u+ϕu=g(u) & \hbox{in ${\mathbb{R}}^3$,} \varepsilon^{2t}(-Δ)^tϕ=u^2,\,\, u>0& \hbox{in ${\mathbb{R}}^3$,} \end{array} \right. \end{equation*} where $s,t\in (0,1)$, $\epsilon >0$ is a small parameter. Under some local assumptions on $V(x)$ and suitable assumptions on the nonlinearity $g$, we construct a family of positive solutions $u_{\epsilon }\in H_{\epsilon }$ which concentrates around the global minima of $V(x)$ as $\epsilon \to 0$.