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Asymptotic Expansions of Solutions of the Yamabe Equation and the σ k ‐Yamabe Equation near Isolated Singular Points
Author(s) -
Han Qing,
Li Xiaoxiao,
Li Yichao
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21943
Subject(s) - mathematics , yamabe flow , degenerate energy levels , mathematical analysis , elliptic curve , mathematical physics , geometry , physics , scalar curvature , curvature , quantum mechanics , sectional curvature
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