Boundary concentration of peak solutions for fractional Schrödinger-Poisson system

2022/01/17 by Shengbing Deng, Deng, Shengbing, Xingliang Tian +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2201.06449

openalex publication_date 2022/01/17 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

The goal of this paper is to study the existence of peak solutions for the following fractional Schrödinger-Poisson system: \ \arraycolsep=1.5pt ε^2s(-Δ)^su+u+ϕu=u^p, · amp; in Ω,
(-Δ)^sϕ=u², · amp; in Ω,
u=ϕ=0, · amp; in ℝ^N∖ Ω, . where s∈(0,1), N>2s, p∈ (1,(N+2s)/(N-2s)), Ω is a bounded domain in ℝ^N with Lipschitz boundary, and (-Δ)^s is the fractional Laplacian operator, ε is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution (u_ε,ϕ_ε) such that the peak of u_ε is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance ε^2/3 away from the boundary ∂Ω as ε tends to 0, some new estimates and analytic technique are used.

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