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

Abstract

The anisotropic Ginzburg–Landau system

$$\begin{aligned} \Delta u+\delta \nabla ({{\,\mathrm{div}\,}}u) +\delta {{\,\mathrm{curl}\,}}^*({{\,\mathrm{curl}\,}}u)=(|u|^2-1) u, \end{aligned}$$

for $u:\mathbb R^2\rightarrow \mathbb R^2$ and $\delta \in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\rightarrow 1$ and u has a prescribed topological degree $d\le -1$ as $|x|\rightarrow \infty$, 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.