Remarks on nontopological solutions in the self-dual Chern-Simons gauged $O(3)$ sigma models

摘要

In this paper, we prove the existence of nontopological solutions to the self-dual equations arising from the Chern-Simons gauged $O(3)$ sigma models. The property of solutions depends on a parameter $au in [-1,1]$ appearing in the nonlinear term. The case $au=1$ lies on the borderline for the existence of solutions in the previous results cite{CH13R2, CH11, CHLLi}. We prove the existence of solutions in this case when there are only vortex points. Moreover, if $-1le au<1$, we establish solutions which are perturbed from the solutions of singular Liouville equations.