A supercritical nonlocal Neumann problem involving non-homogeneous fractional Laplacian

TL;DR

This paper investigates the existence of positive, non-decreasing radial solutions for a nonlocal fractional g-Laplace problem with Neumann boundary conditions, using advanced variational methods in fractional Orlicz-Sobolev spaces.

Contribution

It introduces a novel approach combining fractional Orlicz-Sobolev space properties with variational principles to handle nonlocal problems lacking compactness.

Findings

Existence of positive non-decreasing radial solutions established.

Development of a variational framework for nonlocal fractional g-Laplace problems.

Application of fractional Orlicz-Sobolev space techniques to nonlocal PDEs.

Abstract

In this paper, we study the existence of positive non-decreasing radial solutions of a nonlocal non-standard growth problem ruled by the fractional $g$-Laplace operator with exterior Neumann condition. Our argument exploits some properties of fractional Orlicz-Sobolev spaces combined with a variational principle for nonsmooth functionals, which allows to deal with problems lacking compactness.