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 ) s u ( x ) ≔ p.v ∫ R n g u ( x ) − u ( y ) | x − y | s d y | 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.