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

TL;DR

This paper investigates the asymptotic behavior of solutions to a nonlinear PDE modeling the Gauged Sigma model for Heisenberg ferromagnets, especially near critical parameter values, revealing detailed solution growth at infinity.

Contribution

It establishes the existence of solution sequences with specific asymptotic behaviors and provides estimates in extremal parameter regimes, advancing understanding of the model's solution structure.

Findings

Solutions grow like -2πβ ln|x| at infinity

Existence of solutions with free parameter β in (2, 2(N-M))

Asymptotic estimates near extremal β values

Abstract

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 $N-M>1$. 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 $2(N-M)$.