On steady distributions of self-attracting clusters under friction and fluctuations

We consider the Vlasov-Fokker-Planek equation with a Newtonian, attracting potential and study its stationary solutions, given by the generalized Lane-Emden equation. In a two-dimensional domain we obtain the existence of a critical mass beyond which the system may admit a gravitational collapse. For a one-dimensional model we prove some results on existence, uniqueness, stability and symmetry-breaking of stationary solutions.