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

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: \[ (-Δ_g)^su(x):=\textrm{p.v.}\int_{\mathbb{R}^n}g\left(\frac{u(x)-u(y)}{|x-y|^s}\right)\frac{dy}{|x-y|^{n+s}}, \] 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.

References in corpus (1)

• A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions