  We study the question of existence and uniqueness for the finite temperature
Kohn-Sham equations. For finite volumes, a unique soluion is shown to exists if
the effective potential satisfies a set of general conditions and the coupling
constant is smaller than a certain value. For periodic background potentials,
this value is proven to be volume independent. In this case, the finite volume
solutions are shown to converge as the thermodynamic limit is considered. The
local density approximation is shown to satisfy the general conditions
mentioned above.
