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

Abstract

AbstractWe study asymptotic behaviors of positive solutions to the Yamabe equation and the σk‐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 σk‐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 σk‐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