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 $ (-\triangle)_p^s u\in C^1(\mathbb{R}^n) $ for smooth functions $ u $.