Sharp quantitative estimates of Struwe’s decomposition

In a seminal work, Struwe proved that if 0≤u∈H˙1(Rn) and Γ(u):=‖Δu+u n+2 n−2‖H−1→0, then dist(u,T)→0, where T denotes the manifold of sums of Aubin–Talenti bubbles and dist(u,T) denotes the H˙1(Rn)-distance of u from T. Ciraolo, Figalli, and Maggi obtained the first quantitative version of Struwe’s decomposition with one bubble in all dimensions, namely dist(u,T)≤CΓ(u). For two or more bubbles, Figalli and Glaudo showed a striking dimensional dependent quantitative estimate, namely dist(u,T)≤CΓ(u) when 3≤n≤5, while this is false for n≥6. In this article, we show that dist(u,T)≤C Γ(u)|logΓ(u)|1 2ifn=6,|Γ(u)| n+2 2(n−2)ifn≥7. Furthermore, we show that this inequality is sharp.