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

The goal of this paper is to study the existence of peak solutions for the following fractional Schrödinger-Poisson system:    ε2s(−∆)su+ u+ φu = up, in Ω, (−∆)sφ = u2, in Ω, u = φ = 0, in RN \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 (−∆)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 (uε, φε) such that the peak of uε is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance ε2/3 away from the boundary ∂Ω as ε tends to 0, some new estimates and analytic technique are used.