Let D = Omega\(omega) over bar subset of R-2 be a smooth annular type domain. We consider the simplified Ginzburg-Landau energy E-epsilon (u) = 1/2 integral (D) vertical bar del u vertical bar(2) + 1/4s(2) integral (D) (1 - vertical bar u vertical bar(2))(2) where u : D -> C, and look for minimizers of E-epsilon with prescribed degrees deg (u, partial derivative Omega) = p, deg (u, partial derivative 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 domains with large capacity ( corresponding to thin annulus). We also prove non-existence of minimizers of E-epsilon, for large epsilon, if p not equal q, pq > 0 and if D is a circular annulus with large capacity. Our approach relies on similar results obtained for the Dirichlet energy E-infinity (u) = 1/2 integral (D) vertical bar del u vertical bar(2), on a previous existence result obtained by Berlyand and Golovaty and on a technique developed by Misiats.