Asymptotic behaviors of governing equation of gauged sigma model for Heisenberg ferromagnet. (July 2020) - Chen, Huyuan; Zhou, Feng
Abstract: In this note, we study weak solutions of equation (0.1) Δ u = 4 e u 1 + e u − 4 π ∑ i = 1 N δ p i + 4 π ∑ j = 1 M δ q j in R 2, 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, …, N, (resp. q j, j = 1, …, 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 β having behaviors as − β ln | x | + O ( 1 ) at infinity with a free parameter β ∈ ( 2, 2 ( N − M ) ), and our concern in this paper is to study the asymptotic behavior of b β as β approaching the extremal values 2 and 2 ( N − M ) .