  A central focus of Ginzburg-Landau theory is the understanding and
characterization of vortex configurations. On a bounded domain $\Omega\subseteq
\mathbb{R}^2,$ global minimizers, and critical states in general, of the
corresponding energy functional have been studied thoroughly in the limit
$\epsilon\to 0,$ where $\epsilon>0$ is the inverse of the Ginzburg-Landau
parameter. The presence of an applied magnetic field of strength $h_{ex}\gg 1$
makes possible the existence of stable vortex states. A notable open problem is
whether there are solutions of the Ginzburg-Landau equation for any number of
vortices below $ h_{ex} |\Omega| /2 \pi,$ for external fields of up to
super-heating field strength. The best earlier partial results give, for every
$0<c<1,$ and $K>0,$ the existence of local minimizers of the Ginzburg-Landau
functional with a prescribed number of vortices in the range $1 \leq N \leq
\min \{ K | \log \epsilon |, c ( h_{ex} |\Omega| /2 \pi ) \}$ and for values of
$1\ll_\epsilon h_{ex}$ smaller than a power of the Ginzburg-Landau parameter.
  In this paper, we prove that there are constants $K_1, \alpha>0$ such that
given natural numbers satisfying \[1\leq N \leq
\frac{h_{ex}}{2\pi}(|\Omega|-h_{ex}^{-1/4}),\] local minimizers of the
Ginzburg-Landau functional with this many vortices exist, for fields such that
$K_1\leq h_{ex} \leq 1/\epsilon^{\alpha}.$ Our strategy consists in combining:
the minimization over a subset of configurations for which we can obtain a very
precise localization of vortices; expansion of the energy in terms of a
modified vortex interaction energy that allows for a reduction to a potential
theory problem; and a quantitative vortex separation result for admissible
configurations. Our results provide detailed information about the vorticity
and refined asymptotics of the local minimizers that we construct.

Analysis of PDEs