On Hopf's Lemma for sign-changing supersolutions to fractional Laplacian equations

In this paper we investigate the validity of Hopf’s Lemma for a (possibly sign-changing) function $u\in H_0^s(\mathrm{\Omega })$ satisfying $$(-\mathrm{\Delta }{)}^{s}u(x)\ge c(x)u(x)\text{in }\mathrm{\Omega },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is an open, bounded domain, $c\in L^{\mathrm{\infty }}(\mathrm{\Omega })$, and $(-\mathrm{\Delta }{)}^{s}u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf’s Lemma for $u$ at a point $x_0\in \mathrm{\partial }\mathrm{\Omega }$ is essentially equivalent to the validity of Hopf’s Lemma for the Caffarelli-Silvestre extension of $u$ at the point $(x_0,0)\in {\mathbb{R}}^N\times {\mathbb{R}}^{+}$. We also provide a slightly more precise characterization of a dichotomy result stated in a recent paper by Dipierro, Soave and Valdinoci.