A p(x)-version of Diaz-Saa Inequality and some applications

TL;DR

This paper extends the Diaz-Saa inequality to p(x)-Laplacian operators, enabling new results on solution uniqueness and comparison principles for anisotropic quasilinear elliptic equations.

Contribution

It introduces a novel p(x)-version of the Diaz-Saa inequality, facilitating advanced analysis of anisotropic elliptic equations.

Findings

Established new uniqueness results for solutions.

Developed comparison principles for anisotropic equations.

Extended inequality applicable to p(x)-Laplacian operators.

Abstract

The main result of this work is a new extension of the well known inequality by Diaz and Saa which, in our case, involves an anisotropic operator, such as the p(x)-Laplacian. Our present extension of this inequality enables us to establish several new results on the uniqueness of solutions and comparison principles for some anisotropic quasilinear elliptic equations. Our proofs take advantage of certain convexity properties of the energy functional associated with the p(x)-Laplacian.