Asymptotic behaviors of governing equation of gauged sigma model for Heisenberg ferromagnet. (English) Zbl 1434.35226
Summary: In this note, we study weak solutions of equation \[\Delta u = \frac{ 4 e^u}{ 1 + e^u} - 4 \pi \sum_{i = 1}^N \delta_{p_i} + 4 \pi \sum_{j = 1}^M \delta_{q_j} \quad \text{in} \mathbb{R}^2,\eqno{(0.1)}\] where \(\{ \delta_{p_i} \}_{i = 1}^N\) (resp. \(\{ \delta_{q_j} \}_{j = 1}^M\)) are Dirac masses concentrated at the points \(p_i, i = 1, \dots, N\), (resp. \(q_j\), \(j = 1, \dots, M)\) and \(N - M > 1\). Eq. (0.1) represents a governing equation of gauged sigma model for Heisenberg ferromagnet. We show that it has a sequence of solutions \(u_\beta\) having behaviors as \(- \beta \ln | x | + O (1 )\) at infinity with a free parameter \(\beta \in (2, 2 (N - M) )\), and our concern in this paper is to study the asymptotic behavior of \(b_\beta\) as \(\beta\) approaching the extremal values 2 and \(2 (N - M )\).
MSC:
| 35Q82 | PDEs in connection with statistical mechanics |
| 82D45 | Statistical mechanics of ferroelectrics |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
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