  Let $\mathcal{D} =\Omega \setminus\bar{\omega} \subset \mathbb{R}^2$ be a
smooth annular type domain. We consider the simplified Ginzburg-Landau energy
$E_\epsilon(u)=\frac{1}{2}\int_{\mathcal{D}} |\nabla u|^2
+\frac{1}{4\epsilon^2}\int_{\mathcal{D}} (1-|u|^2)^2$, where $u: \mathcal{D}
\rightarrow \mathbb{C}$, and look for minimizers of $E_\epsilon$ with
prescribed degrees $deg(u,\partial \Omega)=p$, $deg(u,\partial \omega)=q$ on
the boundaries of the domain. For large $\epsilon$ and for balanced degrees,
i.e., $p=q$, we obtain existence of minimizers for {\it thin} domain. We also
prove non-existence of minimizers of $E_\epsilon$, for large $\epsilon$, in the
case $p\neq q$, $pq>0$ and $\mathcal{D}$ is a circular annulus with large
capacity (corresponding to "thin" annulus). Our approach relies on similar
results obtained for the Dirichlet energy
$E_\infty(u)=\frac{1}{2}\int_{\mathcal{D}}|\nabla u|^2$, the existence result
obtained by Berlyand and Golovaty and on a technique developed by Misiats.
