# A Paneitz-Branson type equation with Neumann boundary conditions

## TL;DR

This paper investigates the existence and non-existence of solutions to a fourth-order elliptic equation with Neumann boundary conditions, identifying thresholds for when solutions are constant or non-constant based on Sobolev inequality constants.

## Contribution

It establishes new thresholds for solution rigidity and non-rigidity in a critical Sobolev inequality problem with Neumann boundary conditions, using asymptotic Rayleigh quotient estimates.

## Key findings

- Non-rigidity of solutions above a certain threshold
- Rigidity of solutions below another threshold
- Identification of thresholds for solution behavior

## Abstract

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely we prove that the best constant is achieved by a non-constant solution of the associated fourth-order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.