Asymptotic Expansions of Solutions of the Yamabe Equation and the σ k ‐Yamabe Equation near Isolated Singular Points

We study asymptotic behaviors of positive solutions to the Yamabe equation and the σ 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 σ 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 ≤  k  ≤  n /2 and become degenerate for n /2 <  k  ≤  n . In a significant portion of the paper, we establish a degree 1 expansion for the σ k ‐Yamabe equation for n /2 <  k  <  n , generalizing a similar result for k  = 1 by Korevaar, Mazzeo, Pacard, and Schoen and for 2 ≤  k  ≤  n /2 by Han, Li, and Teixeira. © 2020 Wiley Periodicals LLC