Asymptotic behavior of solutions to the \(\sigma_{k}\)-Yamabe equation near isolated singularities. (English) Zbl 1211.53064
The results of this interesting paper generalize earlier pioneering work on the classical Yamabe equation by Caffarelli, Gidas and Spruck. The authors study the \(\sigma_k\)-Yamabe equation, a conformally invariant type equation generalizing the Yamabe equation. They prove that an admissible solution with an isolated singularity at \(0 \in \mathbb R^n\) to the \(\sigma_k\)-Yamabe equation is asymptotic to a radial solution to the same equation on \(\mathbb R^n \setminus \{0\}.\) In extending the work of Caffarelli et al., the authors 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.
Reviewer: Maria Aparecida Soares Ruas (São Carlos)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35A24 | Methods of ordinary differential equations applied to PDEs |
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