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 nonlin-earity. 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 [9] where the decay rate (1 + |x|) − 12 is assumed. Key words: Vlasov-Poisson-Boltzmann system; stationary solutions; nonlinear elliptic equa-tion. 1