Bubbling solutions for nonlocal elliptic problems

We investigate bubbling solutions for the nonlocal equation $$A_{\mathrm{\Omega }}^{s}u=u^p,u>0in\mathrm{\Omega },$$ under homogeneous Dirichlet conditions, where $\mathrm{\Omega }$ is a bounded and smooth domain. The operator $A_{\mathrm{\Omega }}^s$ 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 $\mathrm{\partial }\mathrm{\Omega }$ and for the restricted fractional Laplacian, we prescribe $u=0$ on ${\mathbb{R}}^n\setminus \mathrm{\Omega }$. We construct solutions when the exponent $p=(n+2s)\mathrm{/}(n-2s)\pm \epsilon$ 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