On the sharp stability of critical points of the Sobolev inequality

Abstract

Given $n\ge 3$, consider the critical elliptic equation $\Delta u+u^{2^{\ast }-1}=0$ in ${\mathbb{R}}^n$ with $u>0$. This equation corresponds to the Euler-Lagrange equation induced by the Sobolev embedding $H^1({\mathbb{R}}^n)\hookrightarrow L^{2^{\ast }}({\mathbb{R}}^n)$, 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, this statement is ``stable up to bubbling'': if $u:{\mathbb{R}}^n\to (0,\infty )$ almost solves $\Delta u+u^{2^{\ast }-1}=0$ then $u$ is (nonquantitatively) close in the $H^1({\mathbb{R}}^n)$-norm to a sum of weakly-interacting Talenti bubbles. More precisely, if $\delta (u)$ denotes the $H^1({\mathbb{R}}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles, Struwe proved that $\delta (u)\to 0$ as $\Vert \Delta u+u^{2^{\ast }-1}{\Vert }_{H^{-1}}\to 0$. 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 $\Vert \nabla u{\Vert }_{L^2}$ (that controls the number of bubbles) it holds $\delta (u)\lesssim \Vert \Delta u+u^{2^{\ast }-1}{\Vert }_{H^{-1}}$. A recent paper by the first author together with Ciraolo and Maggi 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 $3\le n\le 5$ while it is false for $n\ge 6$. 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 $n$.

arXiv comment42 pages

References in corpus (2)

• Alexandrov's theorem revisited
• Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals

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