The asymptotic behavior of Chern-Simons Vortices for Gudnason Model

We consider an elliptic system arising from a supersymmetric gauge field theory. In this paper, we complete to classify all possible solutions according to their asymptotic behavior under a weak coupling effect. Interestingly, it turns out that one of components does not follow the feature of condensate solutions for the classical Chern-Simons-Higgs model. Moreover, in order to prove the concentration property of blow up component, we need to improve the convergence rate and the gradient estimation for the other component, which converges to a constant. We expect that this result would provide an insight for the study of general elliptic system problems, which are even neither cooperative nor competitive.