Steady states of the Vlasov-Poisson-Fokker-Planck system

The form of steady state solutions to the Vlasov-Poisson-Fokker-Planck system is known from the works of Dressler and others. In these papers an external potential is present which tends to infinity as |x| → ∞. It is shown here that this assumption is needed to obtain nontrivial steady states. This is achieved by showing that for a given nonnegative background density satisfying certain integrability conditions, only the trivial solution is possible. This result is sharp and exactly matches the known existence criteria of F. Bouchut and J. Dolbeault (Differential Integral Equations 8, 1995, 487-514) and others. These steady states are solutions to a nonlinear elliptic equation with an exponential nonlinearity. For a given background density which is asymptotically constant, it is directly shown by elementary means that this nonlinear elliptic equation possesses a smooth and uniquely determined global solution.