Maximum principles, Liouville theorem and symmetry results for the fractional $g-$Laplacian

TL;DR

This paper develops maximum principles, Liouville theorems, and symmetry results for the fractional g-Laplacian, a non-local operator in fractional Orlicz-Sobolev spaces, marking the first such results in this setting.

Contribution

It introduces the first maximum principles and symmetry results for the fractional g-Laplacian in fractional Orlicz-Sobolev spaces, expanding the theoretical understanding of these operators.

Findings

Established maximum principles for fractional g-Laplacian.

Proved Liouville type theorems for solutions.

Derived symmetry results and discussed extensions.

Abstract

We study different maximum principles for non-local non-linear operators with non-standard growth that arise naturally in the context of fractional Orlicz-Sobolev spaces and whose most notable representative is the fractional $g-$Laplacian: \[ (-\Delta_g)^su(x):=\textrm{p.v.}\int_{\mathbb{R}^n}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{n+s}}, \] being $g$ the derivative of a Young function. We further derive qualitative properties of solutions such as a Liouville type theorem and symmetry results and present several possible extensions and some interesting open questions. These are the first results of this type proved in this setting.