  We construct local minimizers to the Ginzburg-Landau functional of
superconductivity whose number of vortices N is prescribed and blows up as the
parameter epsilon, inverse of the Ginzburg-Landau parameter kappa, tends to
zero. We treat the case of N as large as log epsilon, and a wide range of
intensity of external magnetic field. The vortices of our solutions arrange
themselves with uniform density over a subregion of the domain bounded by a
"free boundary" determined via an obstacle problem, and asymptotically tend to
minimize the "Coulombian renormalized energy" W introduced in [14].
  The method, inspired by [22], consists in minimizing the energy over a
suitable subset of the functional space, and in showing that the minimum is
achieved in the interior of the subset. It also relies heavily on refined
asymptotic estimates for the Ginzburg-Landau energy obtained in [14].
