Nonlocal problems with Neumann and Robin boundary condition in fractional Musielak-Sobolev spaces

TL;DR

This paper investigates nonlocal fractional Laplacian problems with Neumann and Robin boundary conditions within Musielak-Sobolev spaces, establishing foundational properties and proving the existence of weak solutions using variational methods.

Contribution

It introduces new properties of the fractional Neumann derivative and develops the theory of Musielak-Sobolev spaces for nonlocal problems, proving existence results with variational techniques.

Findings

Established properties of the fractional Neumann derivative.

Proved the existence of weak solutions for nonlocal boundary value problems.

Extended variational methods to Musielak-Sobolev space settings.

Abstract

In this paper, we develop some properties of the $a_{x,y}(.)$-Neumann derivative for the fractional $a_{x,y}(.)$-Laplacian operator. Therefore we prove the basic proprieties of the correspondent function spaces. In the second part of this paper, by means of Ekeland's variational principal and direct variational approach, we prove the existence of weak solutions for a nonlocal problem with nonhomogeneous Neumann and Robin boundary condition.