论文标题
对高级Brezis-Nirenberg问题的积极解决方案的独特性
Uniqueness of positive solutions to the higher order Brezis-Nirenberg problem
论文作者
论文摘要
在本文中，我们研究了Navier边界条件下的高阶Brezis-nirenberg问题\ be \ be \ babel {eq} \ begin {cases}（-Δ） u =-ΔU= \ cdots =（ - δ）^{m-1} u = 0＆\ text {on} \，\ \ partialω，\ end {cases} \ ee，其中$ω$是$ \ mathbb {r} \ Mathbb {n} _ {+} $，$ \ VAREPSILON \ in（0，λ_{1}）$，$λ_{1} $是$（ - δ）^{M}^{m}^{m navier eigenvalue for $Δ我们证明，如果$ \ varepsilon $接近$λ_1$或$ \ varepsilon $接近0和$ω$，则\ eqref {eq}的解决方案是唯一的，并且可以满足某些对称性假设。该证明主要基于我们以前关于爆炸分析和紧凑性结果的较高质量关键椭圆方程的结果以及解决方案对\ eqref {eq}的渐近行为的结果。
In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be\label{eq} \begin{cases} (-Δ)^m u=\varepsilon u+u^{p} & \text { in }\, Ω, \\ u>0 & \text { in }\, Ω, \\ u=-Δu=\cdots=(-Δ)^{m-1} u=0 & \text { on }\, \partial Ω, \end{cases} \ee where $Ω$ is a strictly convex smooth bounded domain in $\mathbb{R}^n$ with $n \geq 4m$, $m \in \mathbb{N}_{+}$, $\varepsilon\in (0,λ_{1})$, $λ_{1}$ is the first Navier eigenvalue for $(-Δ)^{m}$ in $Ω$, and $p=\frac{n+2m}{n-2m}$. We prove that the solutions of \eqref{eq} are unique if either $\varepsilon$ close to $λ_1$ or $\varepsilon$ close to 0 and $Ω$ 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}.
