Existence of Positive Solutions for Some Dirichlet Problems Associated to Fractional Laplacian in Exterior Domains

Abstract

In this paper, we use tools of potential theory to study the existence of positive continuous solutions for some boundary value problems based on the fractional Laplacian $\left( -\Delta \right) ^{\alpha /2},$ $0\lt\alpha \lt 2,$ $% \ $in an exterior domain $D$ in $ \mathbb{R}^{n}$, $n\geq 3$. Our arguments use properties of an appropriate Kato class of functions $K_{\alpha }^{\infty }\left( D\right) .$