S+ property for divergence operator $-{\rm div}\ a(z, u(z), \nabla u(z))$ with $p(\cdot)$-growth condition and applications

TL;DR

This paper establishes criteria for a divergence operator with variable growth conditions to satisfy the S+ property, supported by examples and an application, advancing the understanding of nonlinear PDE operators.

Contribution

It provides new criteria for the S+ property of divergence operators with p(·)-growth, including examples and an application, extending existing theory.

Findings

Criteria for S+ property under p(·)-growth conditions

Examples illustrating the criteria

Application demonstrating practical relevance

Abstract

We establish a result which provides the criteria for a divergence operator dependent both on the gradient and the function itself, to satisfy the S+ property. The result is complemented by several examples and an application.