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:where s ∈ (0, 1), N > 2s, p ∈ (1, N +2s N −2s ), Ω is a bounded domain in R N with Lipschitz boundary, and (−∆) 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.