We consider the gap equation in superconductivity theory with a positive kernel. Under reasonable conditions on the kernel function, we prove that there exists at most one positive solution. This together with the known existence result gives a fairly complete understanding of the gap equation.
Yi Hong Du. "Uniqueness results for the gap equation in the theory of superconductivity." Differential Integral Equations 8 (2) 309 - 324, 1995. https://doi.org/10.57262/die/1369083471