  We analyze an elliptic equation arising in the study of the gauged O(3) sigma
model with the Chern-Simons term. In this paper, we study the asymptotic
behavior of solutions and apply it to prove the uniqueness of stable solutions.
However, one of the features of this nonlinear equation is the existence of
stable nontopological solutions in $\RN$, which implies the possibility that a
stable solution which blows up at a vortex point exists. To exclude this kind
of blow up behavior is one of the main difficulties which we have to overcome.
