Boundary concentration of peak solutions for fractional Schrödinger-Poisson system

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 $s\in(0,1)$, $N>2s$, $p\in (1,\frac{N+2s}{N-2s})$, $Ω$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, and $(-Δ)^{s}$ is the fractional Laplacian operator, $\varepsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_\varepsilon,ϕ_\varepsilon)$ such that the peak of $u_\varepsilon$ is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance $\varepsilon^{2/3}$ away from the boundary $\partialΩ$ as $\varepsilon$ tends to 0, some new estimates and analytic technique are used.

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