Hopf's lemmas and boundary point results for the fractional $p$-Laplacian

Abstract

In this paper, we consider different versions of the classical Hopf's boundary lemma in the setting of the fractional $p-$Laplacian for $p \geq 2$. We start by providing for a new proof to a Hopf's lemma based on comparison principles. Afterwards, we give a Hopf's result for sign-changing potential describing the behavior of the fractional normal derivative of solutions around boundary points. The main contribution here is that we do not need to impose a global condition on the sign of the solution. Applications of the main results to boundary point lemmas and non-local non-linear overdetermined problems are also provided.