Existence and non-uniqueness of stationary states for the Vlasov-Poisson equation on ${\mathbb{R}}^3$ subject to attractive background charges

We prove the existence of stationary solutions for the density of an infinitely extended plasma interacting with an arbitrary configuration of background charges. Furthermore, we show that the solution cannot be unique if the total charge of the background is attractive. In this case, infinitely many different stationary solutions exist. The non-uniqueness can be explained by the presence of trapped particles orbiting the attractive background charge.