Critical points of a non-linear functional related to the one-dimensional Ginzburg–Landau model of a superconducting–normal–superconducting junction

A non-linear functional is studied, which is the limit of the one-dimensional Ginzburg–Landau model of a superconducting–normal–superconducting junction as the Ginzburg–Landau parameter tends to infinity. It is found that the functional may have one, two or three critical points according to various conditions of related parameters and the exact number of the critical points is obtained in each case. Moreover, the necessary and sufficient conditions for minimizers are also established.