Following our preceding papers [1, 2] concerning semi-infinite superconducting films, we consider new a priori estimates on the exterior magnetic field h=A′(0), when (f, A) is a solution of the corresponding Ginzburg–Landau system. The main new results concern the limit as κ→∞, but we prove also the existence of a finite superheating field. We also discuss recent results [3] concerning the superheating field in the large κ limit, and show how to relate these formal solutions to suitable subsolutions and supersolutions giving the existence of a solution for h<1/√2 and κ large enough. We also analyse the same problem by variational techniques and get the existence of a locally stable solution for h<1/√2 and any κ>0.