Existence of positive solution for a fractional elliptic equation in exterior domain

Abstract

In this paper, we show the existence of positive solutions for nonlinear Schrodinger equation with fractional Laplacian

$${(-\mathrm{\Delta })}^{s}u+\lambda u=|u{|}^{p-2}u\mathrm{in}\mathrm{\Omega },$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an unbounded domain, $\partial \mathrm{\Omega }\ne \varnothing$ is bounded, $\lambda \in {\mathbb{R}}^{+}$, $s\in (0,1)$, $N>2s$ and $2<p<2_s^{⁎}$. Precisely, we show that the problem has at least one positive solutions. We achieved our results by using variational method together with Brouwer theory of degree. Moreover, we also prove a version to the Fractional operator in unbounded domain of the Global Compactness result due to Struwe (see [24]).