Boundary concentration of peak solutions for fractional Schrödinger-Poisson system

Abstract

The goal of this paper is to study the existence of peak solutions for the following fractional Schrödinger-Poisson system: \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} \varepsilon^{2s}(-Δ)^{s}u+u+ϕu=u^p,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] (-Δ)^{s}ϕ=u^2,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] u=ϕ=0,\ \ \ \ &\ \mbox{in}\ \mathbb{R}^N\setminus Ω, \end{array} \right. \end{eqnarray*} where $s\in (0,1)$, $N>2s$, $p\in (1,\frac{N+2s}{N-2s})$, $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with Lipschitz boundary, and $(-\Delta {)}^s$ is the fractional Laplacian operator, $\epsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_{\epsilon },{\varphi }_{\epsilon })$ such that the peak of $u_{\epsilon }$ is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance ${\epsilon }^{2/3}$ away from the boundary $\partial \Omega$ as $\epsilon$ tends to 0, some new estimates and analytic technique are used.