Sub-solutions and a point-wise Hopf's Lemma for Fractional p-Laplacian

Abstract

We prove a Hopf's lemma in the point-wise sense. The essential technique is to prove $(-\Delta)^s_p u(x)$ is uniformly bounded in the unit ball $B_1\subset\mathbb{R}^n$, where $u(x)=(1-|x|^2)^s_{+}$. Also we study the global H\"older continuity of bounded positive solutions for $(-\Delta)^s_p u(x)=f(x,u).$