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.
Categories: math.AP