Asymptotic behaviors of governing equation of Gauged Sigma model for Heisenberg ferromagnet

Abstract

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 $\{{\delta }_{p_i}{\}}_{i=1}^N$ (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 $N-M>1$. This equation 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 $2(N-M)$.

16 pages, Nonlinear Analysis, doi.org/10.1016/j.na.2020.111788

References in corpus (1)

• On non-topological solutions for planar Liouville Systems of Toda-type