Sharp quantitative stability of Struwe's decomposition of the Poincar\'e-Sobolev inequalities on the hyperbolic space: Part I

TL;DR

This paper establishes sharp quantitative stability estimates for Struwe's decomposition of solutions to the Poincaré-Sobolev inequalities on hyperbolic space, revealing a dependence on the exponent p and the dimension n, with new techniques tailored to hyperbolic geometry.

Contribution

It proves a new stability inequality for hyperbolic bubbles under certain bounds, extending previous Euclidean results and highlighting the role of the exponent p in subcritical regimes.

Findings

Stability holds for p > 2 and 3 ≤ n ≤ 5.

Stability fails for p ≤ 2 regardless of dimension.

New estimates on hyperbolic bubble interactions and eigenfunction integrability.

Abstract

A classical result owing to Mancini and Sandeep [Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (2008)] asserts that all positive solutions of the Poincar\'e-Sobolev equation on the hyperbolic space $$ -\Delta_{\mathbb{B}^n} u-\lambda u = |u|^{p-1}u, \quad u\in H^1(\mathbb{B}^n), $$ are unique up to hyperbolic isometries where $n \geq 3,$ $1 < p \leq \frac{n+2}{n-2} $ and $\lambda \leq \frac{(n-1)^2}{4}.$ We prove under certain bounds on $\|\nabla u \|_{L^2(\mathbb{B}^n)}$ the inequality $$ \delta(u) \lesssim \|\Delta_{\mathbb{B}^n} u+ \lambda u + u^{p}\|_{H^{-1}}, $$ holds whenever $p >2$ and hence forcing the dimensional restriction $3 \leq n \leq 5,$ where $\delta(u)$ 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.