Uniqueness of positive solutions to the higher order Brezis-Nirenberg problem

TL;DR

This paper proves the uniqueness of positive solutions to a higher order Brezis-Nirenberg problem under certain boundary conditions, domain symmetries, and parameter ranges, extending understanding of critical elliptic equations.

Contribution

It establishes the uniqueness of solutions for the higher order Brezis-Nirenberg problem under specific conditions, using blow-up analysis and asymptotic behavior techniques.

Findings

Solutions are unique when e9 is close to b5_1 or 0.

Uniqueness holds for domains with certain symmetry properties.

The proof relies on blow-up analysis and compactness results.

Abstract

In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-\Delta)^m u=\varepsilon u+u^{p} & \text { in }\, \Omega, \\ u>0 & \text { in }\, \Omega, \\ u=-\Delta u=\cdots=(-\Delta)^{m-1} u=0 & \text { on }\, \partial \Omega, \end{cases} \ee where $\Omega$ is a strictly convex smooth bounded domain in $\mathbb{R}^n$ with $n \geq 4m$, $m \in \mathbb{N}_{+}$, $\varepsilon\in (0,\lambda_{1})$, $\lambda_{1}$ is the first Navier eigenvalue for $(-\Delta)^{m}$ in $\Omega$, and $p=\frac{n+2m}{n-2m}$. We prove that the solutions of \eqref{eq} are unique if either $\varepsilon$ close to $\lambda_1$ or $\varepsilon$ close to 0 and $\Omega$ satisfies some symmetry assumptions. The proof is mainly based on our previous works about the blow up analysis and compactness result for solutions to higher order critical elliptic equations and the asymptotic behavior of solutions to \eqref{eq}.