Asymptotic behaviors of governing equation of Gauged Sigma model for Heisenberg ferromagnet
Huyuan Chen, Feng Zhou · arXiv (Cornell University) · 2018
In this note, we study weak solutions of equation \begin{equation}\label{eq 00.1} Δu =\frac{4e^u}{1+e^u} -4π\sum^{N}_{i=1}δ_{p_i}+4π\sum^{M}_{j=1}δ_{q_j} \quad{\rm in}\;\; \mathbb{R}^2, \end{equation} where $\{δ_{p_i}\}_{i=1}^N$ (resp. $\{δ_{q_j}\}_{j=1}^M$ ) are Dirac masses concentrated at the points $p_i, i=1,\cdots, N$, (resp. $q_j, i=1,\cdots, M$) %$δ_{p_j}$ is Dirac mass concentrated at the point $p_j$ and $N-M>1$. This equation presents a governing equation of Gauged Sigma model for Heisenberg ferromagnet and we prove that it has a sequence of solutions $u_β$ having behaviors as $-2πβ\ln |x|+O(1)$ at infinity with a free parameter $β\in(2,2(N-M))$, and our concern in this paper is to study the asymptotic behavior's estimates in the extremal case that $β$ near $2$ and $2(N-M)$.