Asymptotic expansions of solutions of the Yamabe equation and the ${\sigma }_k$-Yamabe equation near isolated singular points

Abstract

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 $1\le k\le n/2$ and become degenerate for $n/2<k\le n$. 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 $2\le k\le n/2$ by Han, Li, and Teixeira.