We study the variational problem related to the onset of superconductivity that identifies the transition from the normal state to the superconducting state of a sample in the presence of an applied magnetic field. Our concern is a thin sample whose 2-d cross-section has a corner. In particular, we focus on the quarter-plane. We show a first eigenfunction minimizing the associated Rayleigh quotient exists and decays away from the corner. We also give a rigorous upper bound for the eigenvalue which is related to the critical temperature at which superconductivity emerges.