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

Abstract

In this paper, we consider different versions of the classical Hopf’sboundary lemma in the setting of the fractional p−Laplacian for p ≥ 2. Westart by providing for a new proof to a Hopf’s lemma based on comparisonprinciples. Afterwards, we give a Hopf’s result for sign-changing potential describing the behavior of the fractional normal derivative of solutions aroundboundary points. The main contribution here is that we do not need to imposea global condition on the sign of the solution. Applications of the main resultsto boundary point lemmas and a discussion of non-local non-linear overdetermined problems are also provided.