Bubbling solutions for nonlocal elliptic problems

Abstract

We investigate bubbling solutions for the nonlocal equation \[ A_Ω^s u =u^p,\ u >0 \quad \mbox{in } Ω, \] under homogeneous Dirichlet conditions, where $\Omega$ is a bounded and smooth domain. The operator $A_{\Omega }^s$ 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 $s\in (0,1)$ and the Dirichlet conditions are different: for the spectral fractional Laplacian, we prescribe $u=0$ on $\partial \Omega$ and for the restricted fractional Laplacian, we prescribe $u=0$ on ${\mathbb{R}}^n\setminus \Omega$. We construct solutions when the exponent $p=(n+2s)/(n-2s)\pm \epsilon$ is close to the critical one, concentrating as $\epsilon \to 0$ near critical points of a reduced function involving the Green and Robin functions of the domain