Bubbling solutions for nonlocal elliptic problems

We investigate bubbling solutions for the nonlocal equation AsΩu=up, u>0in Ω, under homogeneous Dirichlet conditions, where Ω is a bounded and smooth domain. The operator AsΩ 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 Rn∖Ω. 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.