Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

TL;DR

This paper proves the existence and multiplicity of solutions for Dirichlet problems involving fractional g-Laplacian operators in fractional Orlicz-Sobolev spaces, using sub- and supersolution methods and variational techniques.

Contribution

It introduces new existence and multiplicity results for fractional g-Laplacian problems in Orlicz-Sobolev spaces, including extremal solutions and multiple sign-changing solutions.

Findings

Existence of solutions via sub- and supersolution methods

Existence of extremal solutions under certain conditions

Multiple solutions including positive, negative, and sign-changing ones

Abstract

In this paper, we first prove the existence of solutions to Dirichlet problems involving the fractional $g$-Laplacian operator and lower order terms by appealing to sub- and supersolution methods. Moreover, we also state the existence of extremal solutions. Afterwards, and under additional assumptions on the lower order structure, we establish by variational techniques the existence of multiple solutions: one positive, one negative and one with non-constant sign.