# General fractional Sobolev Space with variable exponent and applications to nonlocal problems

## TL;DR

This paper introduces a generalized fractional Sobolev space with variable exponents and kernels, explores its mathematical properties, and applies these findings to establish solutions for nonlocal elliptic problems with variable exponents.

## Contribution

It extends fractional Sobolev spaces to include variable exponents and kernels, providing new theoretical insights and applications to nonlocal variable exponent problems.

## Key findings

- Proved properties like completeness, reflexivity, and density of the new space.
- Established embedding theorems into variable exponent Lebesgue spaces.
- Demonstrated existence and uniqueness of solutions for related nonlocal problems.

## Abstract

In this paper, we extend the fractional Sobolev spaces with variable exponents $W^{s,p(x,y)}$ to include the general fractional case $W^{K,p(x,y)}$, where $p$ is a variable exponent, $s\in (0,1)$ and $K$ is a suitable kernel. We are concerned with some qualitative properties of the space $W^{K,p(x,y)}$ (completeness, reflexivity, separability, and density). Moreover, we prove a continuous and compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discuss the existence of a nontrivial solution for a nonlocal $p(x,.)$-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type $\mathcal{L}^{p(x,.)}_K$.