带梯度项的 $m$-Laplace 方程 II：第二临界情形的分类
The $m$-Laplace equation with a gradient term II: classification in the second critical case
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中文总结 AI 辅助
本文分类了带梯度项的 $m$-Laplace 方程在第二临界情形下的正弱解，证明解为常数或显式径向解族，无需全局有界或能量假设。
中文摘要 AI 辅助
设 $1<m<n$ 且 $0<q<m-1$。我们分类 $\mathbb R^n$ 中方程 $-\Delta_m u=u^p|Du|^q$ 的正的、局部有界的弱解，其中 $p=\frac{m-q}{n-m}\left(n+\frac{q}{m-1-q}\right)-1$。每个解要么是常数，要么属于一个显式的径向解族（在平移和缩放意义下）。不需要全局有界性、衰减或有限能量假设。证明基于一个标量微分恒等式和一个在临界点仍然成立的最大值原理。这些导致一个尖锐的梯度界、规范化解的分类，以及对任意整体解的首接触论证。局部正则性和临界点处的行为全程在弱解类中处理；不假设全局 $C^2$ 正则性。
英文摘要
Let $1<m<n$ and $0<q<m-1$. We classify positive, locally bounded weak solutions of \[-Δ_m u=u^p|Du|^q\quad\text{in }\mathbb R^n,\qquad p=\frac{m-q}{n-m}\left(n+\frac{q}{m-1-q}\right)-1.\] Every solution is constant or belongs to an explicit family of radial solutions, up to translation and scaling. No global bound, decay, or finite-energy assumption is required. The proof is based on a scalar differential identity and a maximum principle that remains valid at critical points. These lead to a sharp gradient bound, classification of normalized solutions, and a first-contact argument for arbitrary entire solutions. Local regularity and the behavior at critical points are treated throughout in the weak solution class; global $C^2$ regularity is not assumed.
