Sharp multiscale control for high order nonlinear equations

We analyze the behavior of families $(u_\alpha)_{\alpha>0}$ of solutions to the high-order critical equation $P_\alpha u_\alpha=\Delta_g^k u_\alpha +\hbox{lot}=|u_\alpha|^{2^\star-2}u_\alpha$ on a Riemannian manifold $M$, with a uniform bound on the Dirichlet energy. We prove a sharp pointwise control of the $u_\alpha$'s by a sum of bubbles uniformly with respect to $\alpha\to +\infty$, that is $|u_\alpha|\leq C\Vert u_\infty \Vert_\infty +C\sum_{i=1}^NB_{i,\alpha}$ where $u_\infty \in C^{2k}(M)$ and the $(B_{i,\alpha})_\alpha$, $i=1,...,N$ are explicit standard peaks.