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

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