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

## TL;DR

This paper establishes a point-wise Hopf's lemma for the fractional p-Laplacian and investigates the regularity of solutions to related equations, advancing understanding of boundary behavior and continuity properties.

## Contribution

It introduces a novel point-wise Hopf's lemma for the fractional p-Laplacian and analyzes the global Hölder continuity of solutions to the associated nonlinear equations.

## Key findings

- Proved a point-wise Hopf's lemma for the fractional p-Laplacian.
- Showed uniform boundedness of the fractional p-Laplacian of specific functions.
- Demonstrated global Hölder continuity of bounded positive solutions.

## 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 $(-\Delta)^s_p u(x)=f(x,u).$