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 Dressier and others. In these paper s an external potential is present which tends to infinity as \x\ --> infinity. It is shown here that this assumption is needed to obtain no ntrivial steady states. This is achieved by showing that for a given n onnegative background density satisfying certain integrability conditi ons, only the trivial solution is possible. This result is sharp and e xactly matches the known existence criteria of F. Bouchut and J. Dolbe ault (Differential Integral Equations 8, 1995, 487-514) and others. Th ese steady states are solutions to a nonlinear elliptic equation with an exponential nonlinearity. For a given background density which is a symptotically constant, it is directly shown by elementary means that this nonlinear elliptic equation possesses a smooth and uniquely deter mined global solution. (C) 1996 Academic Press, Inc.