Non-stability of Paneitz-Branson equations in arbitrary dimensions

TL;DR

This paper investigates the stability of a class of Paneitz-Branson equations on compact Riemannian manifolds of dimension five or higher, demonstrating that under certain conditions, these equations are generally unstable.

Contribution

It establishes the non-stability of subcritical Paneitz-Branson equations in arbitrary dimensions under specific assumptions, extending understanding of their stability properties.

Findings

Equations are not stable for all dimensions n ≥ 5.

Existence of a positive nondegenerate solution influences stability.

Stability depends on geometric and dimensional conditions.

Abstract

Let $(M,g)$ be a compact riemannian manifold of dimension $n\geq 5$. We are interested in the stability of a slighly subcritical Paneitz-Branson type equation on $M$. Assuming that there exists a positive nondegenerate solution of the critical equation and under suitable conditions, we prove that this equation is not stable for all $n\geq 5$.