Entire Vortex Solutions of Negative Degree for the Anisotropic Ginzburg–Landau System

Abstract

The anisotropic Ginzburg-Landau system \[ \Delta u+\delta\, \nabla (\mathrm{div}\: u) +\delta\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $\delta\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|\delta| < \delta_0(d)$. Unlike the isotropic case $\delta=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.