Multi-vortex solutions to Ginzburg–Landau equations with external potential

We consider the existence of multi-vortex solutions to the Ginzburg-Landau equations with external potential on ℝ². These equations model equilibrium states of superconductors and stationary states of the U(1) Higgs model of particle physics. In the former case, the external potential models impurities and defects. We show that if the external potential is small enough and the magnetic vortices are widely spaced, then one can pin one or an arbitrary number of vortices in the vicinity of a critical point of the potential. In addition, one can pin an arbitrary number of vortices near infinity if the potential is radially symmetric and of an algebraic order near infinity. Copyright © 2011 Springer-Verlag.