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

Abstract

In this article, we study the quantitative stability of the Poincaré-Sobolev equation on the hyperbolic space($P$)$-{\Delta }_{B^n}u-\lambda u=|u{|}^{p-1}u,u\in H^1(B^n),$ where $n\ge 3$, $1<p\le \frac{n+2}{n-2}$ and $\lambda \le \frac{{(n-1)}^2}{4}$. If we consider the upper half space model for the hyperbolic space, then the solutions to ($P$) have certain equivalence with the cylindrically symmetric solutions to the Hardy-Sobolev-Mazy'a equation on the Euclidean space.

A classical result owing to Mancini and Sandeep [43] asserts that all positive solutions to ($P$) are unique up to hyperbolic isometries, which henceforth will be called the hyperbolic bubbles. In the spirit of Struwe, Bhakta-Sandeep [6] proved the following non-quantitative stability result: if $u_m\ge 0$, and ${\Vert {\Delta }_{B^n}{u}_m+\lambda u_m+u_m^p\Vert }_{H^{-1}}\to 0$, then $\delta \left(u_m\right):=\text{dist}(u_m,M_{\lambda })\to 0$, where $\text{dist}(u_m,M_{\lambda })$ denotes the $H^1$-distance of $u_m$ from the manifold of sums of superpositions of hyperbolic bubbles and (localized) Aubin-Talenti bubbles.

In this article, we study the quantitative stability of Struwe decomposition. We prove under certain bounds on ${\Vert \nabla u\Vert }_{L^2\left(B^n\right)}$ the inequality$\delta \left(u\right)\lesssim {\Vert {\Delta }_{B^n}u+\lambda u+u^p\Vert }_{H^{-1}},$ holds whenever $p>2$ and hence forcing the dimensional restriction $3\le n\le 5$. Moreover, it fails for any $n\ge 3$ and $p\in (1,2]$ and hence the dependence on the exponent p is sharp. In the critical case, our dimensional constraint coincides with the seminal result of Figalli and Glaudo [29], which manifests as the dependence on dimension via the critical exponent $\frac{2n}{n-2}>2$ if and only if $3\le n\le 5$.

We derive several new and novel estimates on the interaction of hyperbolic bubbles and their derivatives. The interaction estimates are sharp and of paramount importance, among several other geometric arguments, to adopt the approach of Figalli and Glaudo in this context. The sharpness of the stability estimates requires improved eigenfunction integrability estimates, which we believe are new in this context.