发表机构
Shanghai Jiao Tong University; Fudan University（上海交通大学; 复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对Lipschitz上 epi 锥中的奇异方程，构造并分类全局解，利用局部数据估计增长率、建立非线性边界哈纳克原理变体，明确解的存在性由指数$γ$和锥频率决定。
AI 中文摘要
我们构造并分类了支撑在一般Lipschitz上 epi 锥中的奇异方程$-Δu=f(X)·u^{-γ}$的所有全局解，其中$f(X)$是满足$0<λ≤f(X)≤Λ$的局部Dini连续函数。全局解的存在性与非存在性仅由方程的指数$γ$和锥的“频率”决定。此外，为分类所有全局解，我们引入了若干新方法：首先，利用局部数据估计全局增长率，进而确立全局解“渐近斜率”的有界性；其次，通过建立Kemper边界哈纳克原理的非线性变体，借助“渐近斜率”的“振荡约化”论证完成所有全局解的分类。
英文摘要
We construct and classify all global solutions to the singular equation $$-Δu=f(X)\cdot u^{-γ}$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<λ\leq f(X)\leqΛ$. The existence and non-existence of a global solution is solely determined by the exponent $γ$ of the equation and the ``frequency" of the cone. Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the ``asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an ``oscillation reduction" argument on the ``asymptotic slope".
