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 ∈ H 0 s ( Ω ) u \in H^s_0(\Omega ) satisfying \[ ( − Δ ) s u ( x ) ≥ c ( x ) u ( x ) in Ω , (-\Delta )^s u(x) \geq c(x)u(x) \quad \text {in }\Omega , \] where Ω ⊂ R N \Omega \subset \mathbb {R}^N ( N ≥ 1 N\geq 1 ) is an open set, c ∈ L ∞ ( Ω ) c \in L^\infty (\Omega ) and ( − Δ ) s u (-\Delta )^s u is the fractional Laplacian of u u . We show that, under suitable assumptions, the validity of Hopf’s Lemma for u u at a point x