σk-Yamabe equations are conformally invariant equations generalizing
the classical Yamabe equation. In an earlier work YanYan Li proved that an
admissible solution with an isolated singularity at 0∈Rn to the
σk-Yamabe equation is asymptotically radially symmetric. In this work
we prove that an admissible solution with an isolated singularity at 0∈Rn to the σk-Yamabe equation is asymptotic to a radial
solution to the same equation on Rn∖{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
σk 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.Comment: 55 page