发表机构
School of Mathematical Sciences, Key Laboratory of MEA(Ministry of Education), Shanghai Key Laboratory of PMMP, East China Normal University（华东师范大学数学科学学院、教育部MEA重点实验室、上海市PMMP重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究双曲空间上半线性椭圆方程正径向奇异解，通过分析局部欧几里得奇异与无穷远双曲动力学的相互作用，在不同临界指数区域构造解并分类，证明超临界情形解的存在唯一性及渐近行为。
AI 中文摘要
我们研究半线性椭圆方程 \begin{align*} \Delta_{\mathbb{H}^N} u+\lambda u+u^p=0 \qquad\text{in }\mathbb{H}^N\setminus\{Q\}, \end{align*} 的正径向解，其中 $N\geq 3$, $p>1$, $\lambda\le \frac{(N-1)^2}{4}$，且 $Q\in\mathbb{H}^N$ 是指定的极点。我们的目的是描述在 $\{Q\}$ 附近的局部欧几里得奇异行为与无穷远处的真正双曲动力学之间的相互作用，以及这种相互作用如何随 Serrin 和 Sobolev 临界指数而变化。对于 $\frac{N}{N-2}\le p<\frac{N+2}{N-2}$，我们构造一族正径向奇异解，在无穷远处选择快速指数模式。在极点处，当 $p=\frac{N}{N-2}$ 时，这些解表现出对数修正的基本解轮廓；当 $\frac{N}{N-2}<p<\frac{N+2}{N-2}$ 时，表现出标准的幂律轮廓。在共形不变对 $(p,\lambda)=(\frac{N+2}{N-2},\frac{N(N-2)}{4})$ 处，我们获得一个显式奇异解和非恒定的 Fowler 型解，并通过相关自治方程的正周期轨道对所有具有不可去奇异点的正径向解进行分类。在超临界区域 $p>\frac{N+2}{N-2}$ 且 $\lambda\le\frac{ N(N-2)}{4}$ 时，我们证明了全局正径向奇异解的存在性和唯一性，推导了其在极点附近的两项局部渐近展开，并建立了其在无穷远处行为的尖锐三分法。
英文摘要
We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation \begin{align*} Δ_{\mathbb{H}^N} u+λu+u^p=0 \qquad\text{in }\mathbb{H}^N\setminus\{Q\}, \end{align*} where $N\geq 3$, $p>1$, $λ\le \frac{(N-1)^2}{4}$, and $Q\in\mathbb{H}^N$ is a prescribed pole. Our purpose is to describe how the locally Euclidean singular behavior near ${\{Q}\}$ interacts with the genuinely hyperbolic dynamics at infinity, and how this interaction changes across the Serrin and Sobolev critical exponents. For $\frac{N}{N-2}\le p<\frac{N+2}{N-2}$, we construct a family of positive radial singular solutions selecting the fast exponential mode at infinity. At the pole, these solutions exhibit the logarithmically corrected fundamental-solution profile when $p=\frac{N}{N-2}$, and the standard power-law profile when $\frac{N}{N-2}<p<\frac{N+2}{N-2}.$ At the conformally invariant pair $(p,λ)=(\frac{N+2}{N-2},\frac{N(N-2)}{4})$, we obtain an explicit singular solution and nonconstant Fowler-type solutions, and we classify all positive radial solutions with a nonremovable singularity via the positive periodic orbits of an associated autonomous equation. In the supercritical regime $p>\frac{N+2}{N-2}$ with $λ\le\frac{ N(N-2)}{4}$, we prove existence and uniqueness of the global positive radial singular solution, derive its two-term local asymptotic expansion near the pole, and establish a sharp trichotomy for its behavior at infinity.
Comments31 pages
