Given $$n\ge 3$$ n ≥ 3 , consider the critical elliptic equation $$\Delta u + u^{2^*-1}=0$$ Δ u + u 2 ∗ - 1 = 0 in $${\mathbb {R}}^n$$ R n with $$u > 0$$ u > 0 . This equation corresponds to the Euler–Lagrange equation induced by the Sobolev embedding $$H^1({\mathbb {R}}^n)\hookrightarrow L^{2^*}({\mathbb {R}}^n)$$ H 1 ( R n ) ↪ L 2 ∗ ( R n ) , 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 $$u:{\mathbb {R}}^n\rightarrow \left( 0,\,\infty \right) $$ u : R n → 0 , ∞ almost solves $$\Delta u + u^{2^*-1}=0$$ Δ u + u 2 ∗ - 1 = 0 then u is (nonquantitatively) close in the $$H^1({\mathbb {R}}^n)$$ H 1 ( R n ) -norm to a sum of weakly-interacting Talenti bubbles. More precisely, if $$\delta (u)$$ δ ( u ) denotes the $$H^1({\mathbb {R}}^n)$$ H 1 ( R n ) -distance of u from the manifold of sums of Talenti bubbles, Struwe proved that $$\delta (u)\rightarrow 0$$ δ ( u ) → 0 as . 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 (that controls the number of bubbles) it holds that 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 $$3\le n\le 5$$ 3 ≤ n ≤ 5 while it is false for $$n\ge 6$$ 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 .

中文翻译：

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关于索博列夫不等式临界点的急剧稳定性

给定 $$n\ge 3$$ n ≥ 3 ，考虑临界椭圆方程 $$\Delta u + u^{2^*-1}=0$$ Δ u + u 2 ∗ - 1 = 0 in $$ {\mathbb {R}}^n$$ R n 与 $$u > 0$$ u > 0 。这个方程对应于由 Sobolev 嵌入导出的欧拉-拉格朗日方程 $$H^1({\mathbb {R}}^n)\hookrightarrow L^{2^*}({\mathbb {R}}^n) $$ H 1 ( R n ) ↪ L 2 ∗ ( R n ) ，众所周知，解决方案具有唯一的特征，并由所谓的“人才气泡”给出。此外，由于 Struwe (Math Z 187(4):511–517, 1984) 的基本结果，该语句“稳定到冒泡”：if $$u:{\mathbb {R}}^n\ rightarrow \left( 0,\,\infty \right) $$ u : R n → 0 , ∞ 几乎可以解决 $$\Delta u + u^{2^*-1}=0$$ Δ u + u 2 ∗ - 1 = 0 那么 u 在 $$H^1({\mathbb {R}}^n)$$ H 1 ( R n ) - 归一化为弱相互作用的 Talenti 气泡总和。更准确地说，如果 $$\delta (u)$$ δ ( u ) 表示 $$H^1({\mathbb {R}}^n)$$ H 1 ( R n ) - u 到流形的距离对于 Talenti 泡沫的总和，Struwe 证明了 $$\delta (u)\rightarrow 0$$ δ ( u ) → 0 as 。在本文中，我们研究了临界点稳定性的精确定量版本的有效性：更准确地说，我们询问是否在能量的界限（控制气泡数量）下，第一作者最近发表的一篇论文合在一起与 Ciraolo 和 Maggi (Int Math Res Not 2018(21):6780–6797, 2017) 的研究表明，如果 u 仅接近一个气泡，则上述结果为真。在这里我们证明，令我们惊讶的是，只要有至少两个气泡，那么上面的估计对于 $$3\le n\le 5$$ 3 ≤ n ≤ 5 是正确的，而对于 $$n\ge 6$$ n ≥ 6 是错误的。据我们所知，这是定量稳定性估计如此显着地依赖于空间维度的第一种情况，完全改变了维度 n 的某些特定值的行为。