Existence of stationary solutions to the Vlasov–Poisson–Boltzmann system

In this paper, we study the existence of stationary solutions to the Vlasov–Poisson–Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as |x|→∞. In fact, the stationary Vlasov–Poisson–Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln(e+|x|))−α for some α>0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the Vlasov–Poisson–Fokker–Planck system, J. Math. Anal. Appl. 202 (1996) 1058–1075] where the decay rate is assumed.