Sharp quantitative stability estimates for the Brezis-Nirenberg problem

Abstract

We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding H01 (Q) hooked right arrow -> L 2n n-2(Q) in a smooth bounded domain Q subset of Rn(n >= 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 u0 and the linear term lambda u of the Brezis-Nirenb erg equation, bubble formation, and the boundary effect of the domain Q. 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 [22,32,24] and the smooth closed manifold case studied in [16]. Our proof refines and streamlines several arguments from the existing literature while also resolving new analytical difficulties specific to our setting.