Bubbling solutions for nonlocal elliptic problems

Abstract

We investigate bubbling solutions for the nonlocal equation Aω s u = up, u > 0 in ω, under homogeneous Dirichlet conditions, where ω is a bounded and smooth domain. The operator As ω stands for two types of nonlocal operators that we treat in a unified way: either the spectral fractional Laplacian or the restricted fractional Laplacian. In both cases s ∈ (0, 1), and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe u = 0 on ∂ω, and for the restricted fractional Laplacian, we prescribe u = 0 on ℝnω. We construct solutions when the exponent p = (n+2s)/(n-2s)±ϵ is close to the critical one, concentrating as ϵ → 0 near critical points of a reduced function involving the Green and Robin functions of the domain.