On governing equation of Gauged Sigma model for Heisenberg ferromagnet.

In this note, we study weak solutions of equation \begin{equation}\label{eq 00.1} \Delta u =\frac{4e^u}{1+e^u} -4\pi\sum^{N}_{i=1}\delta_{p_i}+4\pi\sum^{M}_{j=1}\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2, \end{equation} where $\{\delta_{p_i}\}_{i=1}^N$ (resp. $\{\delta_{q_j}\}_{j=1}^M$ ) are Dirac masses concentrated at the points $p_i, i=1,\cdots, N$, (resp. $q_j, i=1,\cdots, M$) %$\delta_{p_j}$ is Dirac mass concentrated at the point $p_j$ and $N-M>1$. Here equation (\ref{eq 00.1}) presents a governing equation of Gauged Sigma model for Heisenberg ferromagnet and we prove that it has a sequence of solutions $u_\beta$ having behaviors as $-2\pi\beta \ln |x|+O(1)$ at infinity with a free parameter $\beta\in(2,2(N-M))$, and our concern in this paper is to study the asymptotic behavior's estimates in the extremal case that $\beta$ near $2$ and $2(N-M)$.