Given n≥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.
