Localization of bubbling for high order nonlinear equations

TL;DR

This paper investigates the behavior of solutions to high-order nonlinear equations on manifolds, identifying obstructions to bubble formation and establishing sharp pointwise estimates using Green's function analysis.

Contribution

It provides new obstructions for solution concentration based on Green's function mass and operator differences, along with precise pointwise bounds for solutions.

Findings

Obstructions depend on Green's function mass and operator differences.

Established sharp pointwise bounds for solutions.

Analyzed high-order critical equations on manifolds.

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.