A duality approach to the fractional Laplacian with measure data

Abstract

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$(-\Delta {)}^{s}v=\mu \text{in}{\mathbb{R}}^N,$$ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure whose support is compactly contained in ${\mathbb{R}}^N$, $N\ge 2$, and $-(\Delta {)}^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.