We study asymptotic behaviors of positive solutions to the Yamabe equation and the $\sigma$k-Yamabe equation near isolated singular points and establish expansions up to arbitrary orders. Such results generalize an earlier pioneering work by Caffarelli, Gidas, and Spruck, and a work by Korevaar, Mazzeo, Pacard, and Schoen, on the Yamabe equation, and a work by Han, Li, and Teixeira on the $\sigma_k$-Yamabe equation. The study is based on a combination of classification of global singular solutions and an analysis of linearized operators at these global singular solutions. Such linearized equations are uniformly elliptic near singular points for $1 \leq k \leq n/2$ and become degenerate for $n/2 < k \leq n$. In a significant portion of the paper, we establish a degree 1 expansion for the $\sigma_k$-Yamabe equation for $n/2 < k < n$, generalizing a similar result for $k = 1$ by Korevaar, Mazzeo, Pacard, and Schoen and for $2 \leq k \leq n/2$ by Han, Li, and Teixeira.

中文翻译：

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Yamabe 方程和 σ k -Yamabe 方程在孤立奇异点附近的解的渐近展开

我们研究了 Yamabe 方程和 $\sigma$k-Yamabe 方程在孤立奇异点附近的正解的渐近行为，并建立了任意阶的扩展。这些结果概括了 Caffarelli、Gidas 和 Spruck 的早期开创性工作，以及 Korevaar、Mazzeo、Pacard 和 Schoen 关于 Yamabe 方程的工作，以及 Han、Li 和 Teixeira 在 $\sigma_k$- 上的工作山部方程。该研究基于全局奇异解的分类和对这些全局奇异解的线性化算子的分析的组合。这种线性化方程在 $1 \leq k \leq n/2$ 的奇异点附近是一致的椭圆，并且在 $n/2 < k \leq n$ 时退化。在论文的重要部分，我们为 $\sigma_k$-Yamabe 方程建立了一个 1 次展开式，其中 $n/2 < k < n$，