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 ε2 u + (1 − |u|2)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. © 2006 Elsevier Inc. All rights reserved