高阶非线性方程气泡现象的局部化
Localization of bubbling for high order nonlinear equations
AI总结：
本文研究高阶临界方程解族的气泡集中现象，给出依赖于维数的集中阻碍条件，并通过精确控制含 Hardy 位势的椭圆算子格林函数，建立了气泡解的逐点上界估计。
摘要：
我们分析高阶临界方程 $P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2-ε_α} u_α\hbox{ in }M$ 的解族的渐近逐点行为，这些解在 $H_k^2(M)$ 中满足 $u_α=u_0+B_α+o(1)$，其中 $B=(B_α)_α$ 是一个气泡（Bubble），也称为峰（Peak）。我们给出这种集中现象发生的阻碍条件：取决于维数，它们涉及相关格林函数的质量，或 $P_α$ 与共形不变的 GJMS 算子之间的差异。本分析的核心是证明逐点控制 \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*} 其中 $|u_α(x_α)|=\max_M|u_α|\to +\infty$ 且 $μ_α:=|u_α(x_α)|^{-\frac{2}{n-2k}}$。获得该估计的关键是对涉及 Hardy 位势的椭圆算子的格林函数进行精确控制。
英文摘要：
We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_αu_α=Δ_g^k u_α+\hbox{lot}=|u_α|^{2^\star-2-ε_α} u_α\hbox{ in }M$$ that behave like $$u_α=u_0+B_α+o(1)\hbox{ in }H_k^2(M)$$ where $B=(B_α)_α$ 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 $P_α$ 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 $|u_α(x_α)|=\max_M|u_α|\to +\infty$ and $μ_α:=|u_α(x_α)|^{-\frac{2}{n-2k}}$. The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.
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