  Aiming at a unified treatment of correlation and inhomogeneity effects in
superconductors, Oliveira, Gross and Kohn proposed in 1988 a density functional
theory for the superconducting state. This theory relies on the existence of a
Kohn-Sham scheme, i.e., an auxiliary noninteracting system with the same
electron and anomalous densities of the original superconducting system.
However, the question of noninteracting $v$-representability has never been
properly addressed and the existence of the Kohn-Sham system has always been
assumed without proof. Here, we show that indeed such a noninteracting system
does not exist in at zero temperature. In spite of this result, we also show
that the theory is still able to yield good results, although in the limit of
weakly correlated systems only.
