Classification of non-topological solutions of an elliptic equation arising from self-dual gauged Sigma model

Our purpose in this paper is to classify the non-topological solutions of equations\begin{document}$ -\Delta u +\frac{4e^u}{1+e^u} = 4\pi\sum\limits_{i = 1}^k n_i\delta_{p_i}-4\pi\sum^l\limits_{j = 1}m_j\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2,\;\;\;\;\;\;(E) $\end{document}where \begin{document}$ \{\delta_{p_i}\}_{i = 1}^k $\end{document} (resp. \begin{document}$ \{\delta_{q_j}\}_{j = 1}^l $\end{document} ) are Dirac masses concentrated at the points \begin{document}$ \{p_i\}_{i = 1}^k $\end{document} , (resp. \begin{document}$ \{q_j\}_{j = 1}^l $\end{document} ), \begin{document}$ n_i $\end{document} and \begin{document}$ m_j $\end{document} are positive integers. Denote \begin{document}$ N = \sum^k_{i = 1}n_i $\end{document} and \begin{document}$ M = \sum^l_{j = 1}m_j $\end{document} satisfying that \begin{document}$ N-M>1 $\end{document} . Problem \begin{document}$ (E) $\end{document} arises from gauged sigma models and we first construct an extremal non-topological solution \begin{document}$ u $\end{document} of \begin{document}$ (E) $\end{document} with asymptotic behavior\begin{document}$ u(x) = -2\ln |x|-2\ln\ln|x|+O(1)\quad{\rm as}\quad |x|\to+\infty $\end{document}and with total magnetic flux \begin{document}$ 4\pi (N-M-1) $\end{document} . And then we do the classification for non-topological solutions of \begin{document}$ (E) $\end{document} with finite magnetic flux. This solves a challenging long standing problem. We believe that our approach is novel and applies to other types of equations.