A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian

TL;DR

This paper establishes a Hopf Lemma and a strong minimum principle for weak supersolutions of the fractional p-Laplacian equation, extending classical results to nonlocal, nonlinear operators.

Contribution

It introduces new Hopf Lemma and strong minimum principle results for fractional p-Laplacian equations, a significant extension of classical PDE theory to nonlocal operators.

Findings

Hopf Lemma for fractional p-Laplacian supersolutions

Strong minimum principle for weak supersolutions

Extension of classical PDE principles to nonlocal fractional operators

Abstract

Our propose here is to provide a Hopf Lemma and a strong minimum principle for week supersolutions of \[ (-\Delta_p)^s u= c(x)|u|^{p-2}u \quad \text{ in } \Omega \] where $\Omega$ is an open set of $\mathbb{R}^N,$ $s\in(0,1),$ $p\in(1,+\infty),$ $c\in C(\overline{\Omega})$ and $(-\Delta_p)^s$ is the fractional $p$-Laplacian.