Sharp quantitative estimates of Struwes Decomposition

Suppose $uin dot{H}^1(mathbb{R}^n)$. In a seminal work, Struwe proved that if $ugeq 0$ and $|Delta u+u^{frac{n+2}{n-2}}|_{H^{-1}}:=Gamma(u)to 0$ then $dist(u,mathcal{T})to 0$, where $dist(u,mathcal{T})$ denotes the $dot{H}^1(mathbb{R}^n)$-distance of $u$ from the manifold of sums of Talenti bubbles. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwes decomposition with one bubble in all dimensions, namely $delta (u) leq C Gamma (u)$. For Struwes decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $delta(u)leq C Gamma(u)$ when $3leq nleq 5$ while this is false for $ ngeq 6$. In this paper, we show that [dist (u,mathcal{T})leq Cbegin{cases} Gamma(u)left|log Gamma(u)right|^{frac{1}{2}}quad&text{if }n=6, |Gamma(u)|^{frac{n+2}{2(n-2)}}quad&text{if }ngeq 7.end{cases}] Furthermore, we show that this inequality is sharp.