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

Commun. Math. Anal.
Volume 13, Number 2 (2012), 27 - 55

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

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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≥3. Our arguments use properties of an appropriate Kato class of functions $K_{\alpha }^{\infty }\left( D\right) .$.