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

σ_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 0∈ℝⁿ to the σ_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 ℝⁿ{set minus}{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 σ_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.