  We study the properties of the Ginzburg-Laundau model in the self-dual point
for a two-dimensional finite system . By a numerical calculation we analyze the
solutions of the Euler-Lagrange equations for a cylindrically symmetric ansatz.
We also study the self-dual equations for this case. We find that the minimal
energy configurations are not given by the Bogomol'nyi equations but by
solutions to the Euler Lagrange ones. With a simple approximation scheme we
reproduce the result of the numerical calculation.
