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 $u \in H^s_0(Ω)$ satisfying \[ (-Δ)^s u(x) \geq c(x)u(x) \quad \text{in }Ω,\] where $Ω\subset \mathbb{R}^N$ is an open, bounded domain, $c \in L^\infty(Ω)$, and $(-Δ)^s u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf's Lemma for $u$ at a point $x_0 \in \partial Ω$ 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.