A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications

TL;DR

This paper develops a weighted, anisotropic version of the Caffarelli-Kohn-Nirenberg inequality within variable exponent spaces and applies it to prove the existence of infinitely many solutions for certain boundary value problems.

Contribution

It introduces a novel weighted anisotropic inequality in variable exponent spaces and demonstrates its application to non-homogeneous boundary value problems.

Findings

Established a new weighted anisotropic Caffarelli-Kohn-Nirenberg inequality.

Proved existence of infinitely many solutions for specific non-homogeneous problems.

Applied the inequality with a variant of the fountain theorem to boundary value problems.

Abstract

We present a weighted version of the Caffarelli-Kohn-Nirenberg inequality in the framework of variable exponents. The combination of this inequality with a variant of the fountain theorem, yields the existence of infinitely many solutions for a class of non-homogeneous problems with Dirichlet boundary condition.