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 vertical bar x vertical bar -> infinity. In fact, the stationary Vlasov-Poisson-Boltzmann system can be written into an elliptic equation with exponential nordinearity. Under the assumption on the decay rate being (ln(e + vertical bar x vertical bar))(-alpha) for some alpha > 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 Viasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl. 202 (1996) 1058-1075] where the decay rate (1 + vertical bar x vertical bar)(-1/2) is assumed. (c) 2006 Elsevier Inc. All rights reserved.