Fractional elliptic problem in exterior domains with nonlocal Neumann boundary condition

TL;DR

This paper investigates the existence of solutions for a fractional elliptic equation with a nonlocal Neumann boundary condition in exterior domains, extending the understanding of fractional Laplacian problems with boundary interactions.

Contribution

It introduces a framework for analyzing fractional elliptic problems with nonlocal Neumann boundary conditions in exterior domains, providing new existence results.

Findings

Existence of solutions under certain conditions.

Extension of fractional Laplacian boundary value problems.

Novel treatment of nonlocal Neumann boundary conditions.

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 $$ \mathcal{N}_su(x) = C_{N,s} \int_{\R^N \setminus \Omega} \frac{u(x) - u(y)}{|x-y|^{N+2s}}dy,\;\;x\in {\Omega}. $$