Localization of bubbling for high order nonlinear equations

Abstract

We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_\alpha u_\alpha=\Delta_g^k u_\alpha+\hbox{lot}=|u_\alpha|^{2^\star-2-\epsilon_\alpha} u_\alpha\hbox{ in }M$$ that behave like $$u_\alpha=u_0+B_\alpha+o(1)\hbox{ in }H_k^2(M)$$ 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_\alpha(x)|\leq C\Vert u_0\Vert_\infty^{(2^\star-1)^2}+C\left(\frac{\mu_\alpha^{2}}{\mu_\alpha^{2 }+d_g(x,x_\alpha)^{2 }}\right)^{\frac{n-2k}{2}}\hbox{ for all }x\in M\hbox{ and }\alpha\in\mathbb{N}, \end{equation*} where $|u_\alpha(x_\alpha)|=\max_M|u_\alpha|\to +\infty$ and $\mu_\alpha:=|u_\alpha(x_\alpha)|^{-\frac{2}{n-2k}}$. The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.