On necessary conditions for the Comparison Principle and the Sub and Supersolutions Method for the stationary Kirchhoff Equation

TL;DR

This paper presents a counterexample demonstrating that the Comparison Principle and Sub and Supersolutions Method may fail for the stationary Kirchhoff Equation in certain conditions, clarifying the necessary criteria for their validity.

Contribution

It identifies a specific condition on the nonlocal term M that is necessary and sufficient for the Comparison Principle to hold in nonlocal Kirchhoff problems.

Findings

Counterexample shows failure of Comparison Principle under certain conditions

Identifies a necessary and sufficient condition on M for validity

Clarifies the gap between known conditions guaranteeing or preventing these properties

Abstract

In this paper we propose a counterexample to the validity of the Comparison Principle and of the Sub and Supersolution Method for nonlocal problems like the stationary Kirchhoff Equation. This counterexample shows that in general smooth bounded domains in any dimension, these properties cannot hold true if the nonlinear nonlocal term $M(\|u\|^2)$ is somewhere increasing with respect to the $H_0^1$-norm of the solution. Comparing with existing results, this fills a gap between known conditions on $M$ that guarantee or prevent these properties, and leads to a condition which is necessary and sufficient for the validity of the Comparison Principle.