On the Sharp Stability of Critical Points of the Sobolev Inequality

Abstract

Given n≥ 3 , consider the critical elliptic equation Δu+u2∗-1=0 in Rn with u> 0. This equation corresponds to the Euler–Lagrange equation induced by the Sobolev embedding H1(Rn)↪L2∗(Rn), and it is well-known that the solutions are uniquely characterized and are given by the so-called “Talenti bubbles”. In addition, thanks to a fundamental result by Struwe (Math Z 187(4):511–517, 1984), this statement is “stable up to bubbling”: if u:Rn→(0,∞)almost solves Δu+u2∗-1=0 then u is (nonquantitatively) close in the H1(Rn) -norm to a sum of weakly-interacting Talenti bubbles. More precisely, if δ(u) denotes the H1(Rn) -distance of u from the manifold of sums of Talenti bubbles, Struwe proved that δ(u) → 0 as [InlineEquation not available: see fulltext.]. In this paper we investigate the validity of a sharp quantitative version of the stability for critical points: more precisely, we ask whether under a bound on the energy [InlineEquation not available: see fulltext.] (that controls the number of bubbles) it holds that [Equation not available: see fulltext.]A recent paper by the first author together with Ciraolo and Maggi (Int Math Res Not 2018(21):6780–6797, 2017) shows that the above result is true if u is close to only one bubble. Here we prove, to our surprise, that whenever there are at least two bubbles then the estimate above is true for 3 ≤ n≤ 5 while it is false for n≥ 6. To our knowledge, this is the first situation where quantitative stability estimates depend so strikingly on the dimension of the space, changing completely behavior for some particular value of the dimension n. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.