A Hardy-Littlewood-Sobolev type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent

TL;DR

This paper extends the Hardy-Littlewood-Sobolev inequality to variable exponents and applies it to prove the existence of solutions for certain quasilinear Choquard equations with variable exponent operators.

Contribution

It introduces a variable exponent version of the Hardy-Littlewood-Sobolev inequality and uses it to establish solutions for Choquard equations involving the p(x)-Laplacian.

Findings

Established a Hardy-Littlewood-Sobolev inequality for variable exponents

Proved existence of solutions for variable exponent Choquard equations

Applied variational methods to nonlinear PDEs with variable exponents

Abstract

In this work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the $p(x)$-Laplacian operator.