Localization of bubbling for high order nonlinear equations

We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_{\alpha }{u}_{\alpha }={\mathrm{\Delta }}_g^k{u}_{\alpha }+\text{lot}=\mathrm{\mid }{u}_{\alpha }{\mathrm{\mid }}^{2^{\star }-2-{\epsilon }_{\alpha }}{u}_{\alpha }\text{ in }M$$ that behave like $$u_{\alpha }=u_0+B_{\alpha }+o(1)\text{ in }{H}_k^2(M)$$ where $B=(B_{\alpha }{)}_{\alpha }$ is a Bubble, also called a Peak. We give obstructions for such a concentration to occur: depending on the dimension, they involve the mass of the associated Green’s function or the difference between $P_{\alpha }$ and the conformally invariant GJMS operator. The bulk of this analysis is the proof of the pointwise control \begin{equation*} |u_α(x)|\leq C\Vert u_0\Vert_\infty^{(2^\star-1)^2}+C\left(\frac{μ_α^{2}}{μ_α^{2 }+d_g(x,x_α)^{2 }}\right)^{\frac{n-2k}{2}}\hbox{ for all }x\in M\hbox{ and }α\in$\mathbb{N}$, \end{equation*} where $\mathrm{\mid }{u}_{\alpha }(x_{\alpha })\mathrm{\mid }={\max }_M\mathrm{\mid }{u}_{\alpha }\mathrm{\mid }\to +\mathrm{\infty }$ and ${\mu }_{\alpha }:=\mathrm{\mid }{u}_{\alpha }(x_{\alpha }){\mathrm{\mid }}^{-\frac{2}{n-2k}}$. The key to obtain this estimate is a sharp control of the Green’s function for elliptic operators involving a Hardy potential.