A Hopf lemma and regularity for fractional $p-$Laplacians

Abstract

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-△{)}_p^{s}u\in C^1({\mathbb{R}}^n)$ for smooth functions $u$.

20 pages