论文标题
Struwe分解Poincaré-Sobolev在双曲线空间上的不平等现象的急剧定量稳定性：第一部分
Sharp quantitative stability of Struwe's decomposition of the Poincaré-Sobolev inequalities on the hyperbolic space: Part I
论文作者
论文摘要
由于Mancini和Sandeep的经典结果[Ann。 sc。规范。极好的。比萨CL。科学。 7（2008）]断言，Poincaré-Sobolev方程的所有积极解决方案$$-δ_ {\ Mathbb {\ Mathbb {b}^n} u-λu= | U | U | U |^{p-1} u，\ quad quad quad U \ in h^in h^in h^1（$ n^up s opery with up took with up took with up took wities up took to n op to 3，$ 1 <p \ leq \ frac {n+2} {n-2} $和$λ\ leq \ leq \ frac {（n-1）^2} {4} {4}。$我们在$ \ | \ | \ | \ | \ nabla u \ \ | _ _ {l^2（l^2（l^2）（ \ lyseSim \ |δ_ {\ Mathbb {b}^n} u +λu + U^{p} {p} \ | _ {h^{ - 1}}，$$ 每当$ p> 2 $时，持有尺寸限制$ 3 \ leq n \ leq 5，其中$δ（u）$表示$ h^1 $ us的$ h^1 $距离的$ u $ $ u $从高血压气泡的多数总和中。 Moreover, it fails for any $n \geq 3$ and $p \in (1,2].$ This strengthens the phenomenon observed in the Euclidean case that the (linear) quantitative stability estimate depends only on whether the exponent $p$ is $>2$ or $\leq 2$. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [Arch. Ration.肛门。双曲线空间的等距组，我们认为与欧几里得病例相比，在参数和技术中有一个显着的区别，可以实现我们的主要结果。
A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincaré-Sobolev equation on the hyperbolic space $$ -Δ_{\mathbb{B}^n} u-λu = |u|^{p-1}u, \quad u\in H^1(\mathbb{B}^n), $$ are unique up to hyperbolic isometries where $n \geq 3,$ $1 < p \leq \frac{n+2}{n-2} $ and $λ\leq \frac{(n-1)^2}{4}.$ We prove under certain bounds on $\|\nabla u \|_{L^2(\mathbb{B}^n)}$ the inequality $$ δ(u) \lesssim \|Δ_{\mathbb{B}^n} u+ λu + u^{p}\|_{H^{-1}}, $$ holds whenever $p >2$ and hence forcing the dimensional restriction $3 \leq n \leq 5,$ where $δ(u)$ denotes the $H^1$ distance of $u$ from the manifold of sums of hyperbolic bubbles. Moreover, it fails for any $n \geq 3$ and $p \in (1,2].$ This strengthens the phenomenon observed in the Euclidean case that the (linear) quantitative stability estimate depends only on whether the exponent $p$ is $>2$ or $\leq 2$. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [Arch. Ration. Mech. Anal, 237 (2020)] but we notice a striking dependence on the exponent $p$ in the subcritical regime as well which is not present in the flat case. Our technique is an amalgamation of Figalli and Glaudo's method and builds upon a series of new and novel estimates on the interaction of hyperbolic bubbles and their derivatives and improved eigenfunction integrability estimates. Since the conformal group coincides with the isometry group of the hyperbolic space, we perceive a remarkable distinction in arguments and techniques to achieve our main results compared to that of the Euclidean case.
