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} (-Δ)^su + u &= Q(x) |u|^{p-1}u\;\;\mbox{in}\;\;\R^N \setminus Ω\\ \mathcal{N}_su(x) &= 0\;\;\mbox{in}\;\;Ω, \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N>2s$, $\Omega \subset {\mathbb{R}}^N$ 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 $${\mathcal{N}}_{s}u(x)=C_{N,s}{\int }_{{\mathbb{R}}^N\setminus \Omega }\frac{u(x)-u(y)}{|x-y{|}^{N+2s}}dy,x\in \Omega .$$

In this version we corrected some misprints. The final version this manuscript will be published in Nonlinear Analysis