Sharp quantitative stability estimates for the Brezis-Nirenberg problem

TL;DR

This paper establishes the first quantitative stability estimates for the Brezis-Nirenberg problem on bounded domains, revealing unexpected stability exponents and addressing boundary effects that distinguish it from Euclidean and manifold cases.

Contribution

It provides the first quantitative stability result for the Sobolev inequality on bounded domains, incorporating boundary effects and novel stability exponents.

Findings

First quantitative stability estimate for Sobolev inequality on bounded domains.

Discovery of unexpected stability exponents due to boundary and bubble interactions.

Refinement of analytical techniques to handle boundary effects in stability analysis.

Abstract

We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 3$). To the best of our knowledge, this work presents the first quantitative stability result for the Sobolev inequality on bounded domains. A key discovery is the emergence of unexpected stability exponents in our estimates, which arise from the intricate interaction among the nonnegative solution $u_0$ and the linear term $\lambda u$ of the Brezis--Nirenberg equation, bubble formation, and the boundary effect of the domain $\Omega$. One of the main challenges is to capture the boundary effect quantitatively, a feature that fundamentally distinguishes our setting from the Euclidean case treated in \cite{CFM, FG, DSW} and the smooth closed manifold case studied in \cite{CK}. In addressing a variety of difficulties, our proof refines and streamlines several arguments from the existing literature while also resolving new analytical challenges specific to our setting.