Concentrating bounded states for fractional Schrödinger-Poisson system involving critical Sobolev exponent

Abstract

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

35pages