Sharp quantitative estimates of Struwe's Decomposition

Abstract: Suppose $u\in \dot{H}^1(\mathbb{R}^n)$. In a seminal work, Struwe proved that if $u\geq 0$ 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 Struwe's decomposition with one bubble in all dimensions, namely $\delta (u) \leq C \Gamma (u)$. For Struwe's decomposition with two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $\delta(u)\leq C \Gamma(u)$ when $3\leq n\leq 5$ while this is false for $ n\geq 6$. In this paper, we show that [dist (u,\mathcal{T})\leq C\begin{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 }n\geq 7.\end{cases}] Furthermore, we show that this inequality is sharp.