Hopf's lemmas and boundary behaviour of solutions to the fractional Laplacian in Orlicz-Sobolev spaces

TL;DR

This paper extends Hopf's boundary lemma to nonlocal, nonlinear fractional operators in Orlicz-Sobolev spaces, analyzing boundary behavior of solutions with various potentials.

Contribution

It introduces new boundary estimates for fractional $a$-Laplacian operators in Orlicz-Sobolev spaces, covering sign-changing potentials.

Findings

Boundary behavior characterized for solutions near domain boundary.

Extensions of Hopf's lemma established for nonlocal, nonlinear operators.

Results applicable to problems with sign-changing potentials.

Abstract

In this article we study different extensions of the celebrated Hopf's boundary lemma within the context of a family of nonlocal, nonlinear and nonstandard growth operators. More precisely, we examine the behavior of solutions of the fractional $a-$Laplacian operator near the boundary of a domain satisfying the interior ball condition. Our approach addresses problems involving both constant-sign and sign-changing potentials.