Variational reduction for Ginzburg–Landau vortices

Let Ω be a bounded domain with smooth boundary in R2. We construct non-constant solutions to the complex-valued Ginzburg–Landau equation ε2Δu+(1−2|u|)u=0 in Ω, as ε→0, both under zero Neumann and Dirichlet boundary conditions. We reduce the problem of finding solutions having isolated zeros (vortices) with degrees ±1 to that of finding critical points of a small C1-perturbation of the associated renormalized energy. This reduction yields general existence results for vortex solutions. In particular, for the Neumann problem, we find that if Ω is not simply connected, then for any k⩾1 a solution with exactly k vortices of degree one exists.