  Density-functional theory for superfluid systems is developed in the
framework of the functional renormalization group based on the effective action
formalism. We introduce the effective action for the particle-number and
nonlocal pairing densities and demonstrate that the Hohenberg-Kohn theorem for
superfluid systems is established in terms of the effective action. The flow
equation for the effective action is then derived, where the flow parameter
runs from $0$ to $1$, corresponding to the non-interacting and interacting
systems. From the flow equation and the variational equation that the
equilibrium density satisfies, we obtain the exact expression for the Kohn-Sham
potential generalized to including the pairing potentials. The resultant
Kohn-Sham potential has a nice feature that it expresses the microscopic
formulae of the external, Hartree, pairing, and exchange-correlation terms,
separately. It is shown that our Kohn-Sham potential gives the ground-state
energy of the Hartree-Fock-Bogoliubov theory by neglecting the correlations. An
advantage of our exact formalism lies in the fact that it provides ways to
systematically improve the correlation part.
