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

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}.