Sub-solutions and a point-wise Hopf's lemma for fractional $ p $ -Laplacian
Zaizheng Li, Qidi Zhang · Communications on Pure & Applied Analysis · 2020
We prove a Hopf's lemma in the point-wise sense for fractional $ p $-Laplacian. 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 Hölder continuity of bounded positive solutions for $ (-\Delta)^s_p u(x) = f(x,u). $