The existence of Ginzburg-Landau solutions on the plane by a direct variational method

This paper studies the existence and the minimization problem of the solutions of the Ginzburg-Landau equations in R² coupled with an external magnetic field or a source current. The lack of a suitable Sobolev inequality makes it necessary to consider a variational problem over a special admissible space so that the space norms of the gauge vector fields of a minimization sequence can be controlled by the corresponding energy upper bound and a solution may be obtained as a minimizer of a modified energy of the problem. Asymptotic properties and flux quantization are established for finite-energy solutions. Besides, it is shown that the solutions obtained also minimize the original Ginzburg-Landau energy when the admissible space is properly chosen.