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

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