A note on Hopf's lemma and strong minimum principle for nonlocal equations with non-standard growth

TL;DR

This paper extends Hopf's lemma and the strong minimum principle to nonlocal equations with non-standard growth, specifically in fractional Orlicz-Sobolev spaces, providing new theoretical insights for weak supersolutions.

Contribution

It introduces a Hopf's lemma and strong minimum principle for nonlocal equations with non-standard growth, extending previous results to fractional Orlicz-Sobolev settings.

Findings

Established Hopf's lemma for nonlocal equations with non-standard growth.

Proved strong minimum principle for weak supersolutions in fractional Orlicz-Sobolev spaces.

Extended classical results to a broader nonlocal, non-standard growth context.

Abstract

Let $\Omega\subset \mathbb{R}^n $ be any open set and $u$ be a weak supersolution of $\mathcal{L}u=c(x)g(|u|)\frac{u}{|u|}$ where \[\mathcal{L}u(x)=\text{p.v.} \int_{\mathbb{R}^n} g\left(\frac{|u(x)-u(y)|}{|x-y|^s}\right) \frac{u(x)-u(y)}{|u(x)-u(y)|} K(x,y)\frac{dy}{|x-y|^s}\] and $g=G^{\prime}$ for some Young function $G.$ This note imparts a Hopf's type lemma and strong minimum principle for $u$ when $c(x)$ is continuous in $\bar{\Omega}$ that extend the results of Del Pezzo and Quaas (JDE-2017) in fractional Orlicz-Sobolev setting.