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 $(-\Delta {)}_p^{s}u(x)$ 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 Hölder continuity of bounded positive solutions for $(-\Delta {)}_p^{s}u(x)=f(x,u).$

33 pages