The same class of stationary solutions to some multidimensional kinetic systems with extensive background density

Abstract

In this study, we investigate the existence and uniqueness of the same class of stationary solutions for some kinetic systems comprising the Vlasov–Poisson–Fokker–Planck system, Vlasov–Poisson–Boltzmann system, and Vlasov–Maxwell–Boltzmann system in arbitrary space dimensions N(N≥3). Essentially, the problem can be reduced to solving the problem of a second order elliptic equation with exponential nonlinearity. This result had been proved in three spatial dimensions but the extension to a higher-dimensional setting makes the existence proof nontrivial. In particular, it necessary to highlight that the requirement regarding the background density function is relaxed compared with previous studies in the case of three spatial dimensions.