Some remarks on the comparison principle in Kirchhoff equations

TL;DR

This paper investigates the limitations of the comparison principle in Kirchhoff equations with increasing Kirchhoff functions, providing counterexamples and proposing an alternative sub-supersolution method for certain models.

Contribution

It demonstrates the failure of the comparison principle under specific conditions and introduces a new sub-supersolution approach for Kirchhoff equations.

Findings

Comparison principle fails for increasing Kirchhoff functions

Counterexamples contradict previous assumptions

An alternative sub-supersolution method is proposed

Abstract

In this paper we study the validity of the comparison principle and the sub-supersolution method for Kirchhoff type equations. We show that these principles do not work when the Kirchhoff function is increasing, contradicting some previous results. We give an alternative sub-supersolution method and apply it to some models.