On the sharp stability of critical points of the Sobolev inequality

Given $n\geq 3$, consider the critical elliptic equation $\Delta u + u^{2^*-1}=0$ in $\mathbb R^n$ with $u > 0$. This equation corresponds to the Euler-Lagrange equation induced by the Sobolev embedding $H^1(\mathbb R^n)\hookrightarrow L^{2^*}(\mathbb R^n)$, 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 $\Delta u + u^{2^*-1}=0$ 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 $\lVert\Delta u + u^{2^*-1}\rVert_{H^{-1}}\to 0$. 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 $\lVert\nabla u\rVert_{L^2}$ (that controls the number of bubbles) it holds $\delta(u) \lesssim \lVert\Delta u + u^{2^*-1}\rVert_{H^{-1}}$. 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 $n$.