Sharp multiscale control for high order nonlinear equations

Abstract

We analyze the behavior of families $(u_{\alpha }{)}_{\alpha >0}$ of solutions to the high-order critical equation $P_{\alpha }{u}_{\alpha }={\Delta }_g^k{u}_{\alpha }+lot=|u_{\alpha }{|}^{2^{\star }-2}{u}_{\alpha }$ 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 $|u_{\alpha }|\le C\Vert u_{\infty }{\Vert }_{\infty }+C{\sum }_{i=1}^N{B}_{i,\alpha }$ where $u_{\infty }\in C^{2k}(M)$ and the $(B_{i,\alpha }{)}_{\alpha }$, $i=1,...,N$ are explicit standard peaks.

38 pages