Robin fractional problems with symmetric variable growth

TL;DR

This paper investigates the fractional p(., .)-Laplacian operator with symmetric variable growth, introducing a nonlocal conormal derivative, establishing fundamental properties, and proving the existence of solutions for Robin boundary problems with sign-changing potentials.

Contribution

It introduces a nonlocal conormal derivative for the fractional p(., .)-Laplacian and proves existence results for related boundary value problems with variable growth.

Findings

Defined the nonlocal conormal derivative for the operator

Established a nonlocal divergence theorem

Proved existence of weak solutions for Robin problems

Abstract

In this paper we study the fractional p(., .)-Laplacian and we introduce the corresponding nonlocal conormal derivative for this operator. We prove basic properties of the corresponding function space and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, we prove the existence of weak solutions of corresponding p(., .)-Robin boundary problems with sign-changing potentials by applying variational tools.