  The Kohn-Sham (KS) equations determine, in a self-consistent way, the
particle density of an interacting fermion system at thermal equilibrium. We
consider a situation when the KS equations are known to have a unique solution
at high temperatures and this solution is a uniform particle density. We show
that, at zero temperature, there are stable solutions that are not uniform. We
provide the general principles behind this phenomenon, namely the conditions
when it can be observed and how to construct these non-uniform solutions. Two
concrete examples are provided, including fermions on the sphere which are
shown to crystallize in a structure that resembles the C$_{60}$ molecule.
