  We consider superconductivity in a system with $N$ Fermi surfaces, including
intraband and interband effective electron-electron interactions. The effective
interaction is described by an $N \times N$ matrix whose elements are assumed
to be constant in thin momentum shells around each Fermi surface, giving rise
to $s$-wave superconductivity. Starting with attractive intraband interactions
in all $N$ bands, we show that too strong interband interactions are
detrimental to sustaining $N$ nonzero components of the superconducting order
parameter. We find similar results in systems with repulsive intraband
interactions. The dimensionality reduction of the order-parameter space is
given by the number of nonpositive eigenvalues of the interaction matrix. Using
general models and models for superconducting transition metal dichalcogenides
and iron pnictides frequently employed in the literature, we show that
constraints must be imposed on the order parameter to ensure a lower bound on
the free energy and that subsequent higher-order expansions around the global
minimum are thermodynamically stable. We also demonstrate that similar
considerations are necessary for unconventional pairing symmetries.
