Fractional elliptic problem in exterior domains with nonlocal Neumann boundary condition

Abstract: In this paper we consider the existence of solution for the following class of fractional elliptic problem \begin{equation}\label{00} \left{\begin{aligned} (-\Delta)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus \Omega\ \mathcal{N}su(x) &= 0\;\;\mbox{in}\;\;{\Omega}, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N> 2s$, $\Omega\subset \R^N$ is a bounded set with smooth boundary, $(-\Delta)^s$ denotes the fractional Laplacian operator and $\mathcal{N}_s$ is the nonlocal operator that describes the Neumann boundary condition, which is given by $$ \mathcal{N}_su(x) = C{N,s} \int_{\R^N \setminus \Omega} \frac{u(x) - u(y)}{|x-y|^{N+2s}}dy,\;\;x\in {\Omega}. $$