In a seminal work, Struwe proved that if $0\leq u\in \dot{H}^1(\mathbb{R}^n)$ and $\Gamma(u):=\|\Delta u+u^{\frac{n+2}{n-2}}\|_{H^{-1}}\to 0$ then $dist(u,\mathcal{T})\to 0$, where $\mathcal{T}$ denotes the manifold of sums of Aubin-Talenti bubbles and $dist(u,\mathcal{T})$ denotes the $\dot{H}^1(\mathbb{R}^n)$-distance of $u$ from $\mathcal{T}$. Ciraolo, Figalli and Maggi obtained the first quantitative version of Struwe's decomposition with one bubble in all dimensions, namely $dist (u,\mathcal{T}) \leq C \Gamma (u)$. For two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely $dist(u,\mathcal{T})\leq C \Gamma(u)$ when $3\leq n\leq 5$ while this is false for $ n\geq 6$. In this talk, I will first show how to get a quantitative estimate, essentially a nonlinear inequality in higher dimensions. Afterward, I will show that this inequality is sharp by constructing an example. This is a joint work with Liming Sun (AMSS) and Juncheng Wei (UBC/CUHK).
