Asymptotic behavior of solutions to the σ k -Yamabe equation near isolated singularities

• 成果类型: Article
• 署名作者: Han, Zheng-Chao; Li, YanYan; Teixeira, Eduardo V.
• 署名单位: Rutgers University System; Rutgers University New Brunswick; Universidade Federal do Ceara
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0274-7
• 发表日期: 2010
• 页码: 635-684
• 关键词: nonlinear elliptic-equations; scalar curvature metrics; blow-up phenomena; variational characterization; conformal geometry; schouten tensor; local behavior; compactness; symmetry; EXISTENCE

摘要

sigma (k) -Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In (J. Funct. Anal. 233: 380-425, 2006) YanYan Li proved that an admissible solution with an isolated singularity at 0aae (n) to the sigma (k) -Yamabe equation is asymptotically radially symmetric. In this work we prove that such a solution is asymptotic to a radial solution to the same equation on ae (n) a-{0}. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al., we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and sigma (k) curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.