Boundary concentration of peak solutions for fractional Schr\"{o}dinger-Poisson system

TL;DR

This paper constructs boundary-concentrating peak solutions for a fractional Schr"{o}dinger-Poisson system, revealing solutions that peak near the boundary at a scale of approximately ^{2/3} as the parameter  tends to zero.

Contribution

The paper introduces new estimates and techniques to analyze boundary concentration phenomena for fractional Schr"{o}dinger-Poisson systems, using Lyapunov-Schmidt reduction.

Findings

Existence of single peak solutions near the boundary.

Peak concentration occurs at a distance proportional to ^{2/3} from the boundary.

New analytic methods for boundary concentration analysis.

Abstract

The goal of this paper is to study the existence of peak solutions for the following fractional Schr\"{o}dinger-Poisson system: \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} \varepsilon^{2s}(-\Delta)^{s}u+u+\phi u=u^p,\ \ \ &\ \mbox{in}\ \Omega,\\[2mm] (-\Delta)^{s}\phi=u^2,\ \ \ &\ \mbox{in}\ \Omega,\\[2mm] u=\phi=0,\ \ \ \ &\ \mbox{in}\ \mathbb{R}^N\setminus \Omega, \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, $\varepsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_\varepsilon,\phi_\varepsilon)$ 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 $\partial\Omega$ as $\varepsilon$ tends to 0, some new estimates and analytic technique are used.