A-priori bounds and multiplicity of solutions for nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian

TL;DR

This paper establishes fundamental imbeddings for variable exponent fractional Sobolev spaces and applies these results to prove a-priori bounds and multiple solutions for nonlinear elliptic problems involving the fractional p(x)-Laplacian.

Contribution

It introduces new imbedding results for fractional Sobolev spaces with variable exponents and applies them to analyze nonlinear elliptic equations with the fractional p(x)-Laplacian.

Findings

Derived fundamental imbeddings for fractional Sobolev spaces with variable exponent.

Proved a-priori bounds for solutions of nonlinear elliptic problems.

Established the existence of multiple solutions for these problems.

Abstract

We obtain fundamental imbeddings for the fractional Sobolev space with variable exponent that is a generalization of well-known fractional Sobolev spaces. As an application, we obtain a-priori bounds and multiplicity of solutions to some nonlinear elliptic problems involving the fractional $p(\cdot)$-Laplacian.