  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.

Differential Geometry;Analysis of PDEs