Mathematical analysis of the multiband BCS gap equations in superconductivity

In this paper, we present a mathematical analysis for the phonon-dominated multiband isotropic and anisotropic BCS gap equations at any finite temperature T. We establish the existence of a critical temperature Tc so that, when TTc, the only nonnegative gap solution is the zero solution, representing the normal phase. Furthermore, when T=Tc, we prove that the only gap solution is the zero solution and that the positive gap solution depend on the temperature T