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 (1 + | x |)^{- frac(1, 2)} is assumed. © 2006 Elsevier Inc. All rights reserved.