Localization of bubbling for high order nonlinear equations
数学专题报告
报告题目(Title)： Localization of bubbling for high order nonlinear equations
报告人(Speaker)：Frédéric Robert (Institut Élie Cartan)
地点(Place)：Zoom ID: 965 6529 2221, PWD: 269806
时间(Time)：2025 年 3月 6日 16：00—17：00
邀请人(Inviter)：熊金钢
报告摘要
We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation
\[
P_\alpha{u_\alpha}=\Delta_g^k{u_\alpha} +\text{l.o.t.}=|{u_\alpha}|^{2^\star-2-{\epsilon_\alpha}}{u_\alpha}\text{ in }M
\]
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
\[
|{u_\alpha}(x)|\leq C\|u_0\|_\infty^{(2^\star-1)^2}+C\left(\frac{{\mu_\alpha}^{2}}{{\mu_\alpha}^{2}+d_g(x,{x_\alpha})^{2}}\right)^{\frac{n-2k}{2}}\text{ for all }x\in M\text{ and }\alpha>0,
\]
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.
* This PDE seminar is co-organized with Tianling Jin at HKUST and Juncheng Wei at CUHK. See the seminar webpage: https://www.math.hkust.edu.hk/~tianlingjin/PDEseminar.html
