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$\sigma_k$-Yamabe equations are conformally invariant equations generalizingthe classical Yamabe equation. In an earlier work YanYan Li proved that anadmissible solution with an isolated singularity at $0\in \mathbb R^n$ to the$\sigma_k$-Yamabe equation is asymptotically radially symmetric. In this workwe 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 radialsolution to the same equation on $\mathbb R^n \setminus \{0\}$. These resultsgeneralize earlier pioneering work in this direction on the classical Yamabeequation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelliet al, we formulate and prove a general asymptotic approximation result forsolutions to certain ODEs which include the case for scalar curvature and$\sigma_k$ curvature cases. An alternative proof is also provided usinganalysis of the linearized operators at the radial solutions, along the linesof approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.

Asymptotic behavior of solutions to the $\sigma_k$-Yamabe equation near isolated singularities