  We prove existence of global weak solutions for the Nernst-Planck-Poisson
problem which describes the evolution of concentrations of charged species
$X_1, ..., X_P$ subject to Fickian diffusion and chemical reactions in the
presence of an electrical field, including in particular the Boltzmann
statistics case. In contrast to the existing literature, existence is proved in
any dimension. Moreover, we do not need the assumption $P = 2$ nor the
assumption of equal diffusivities for all $P$ components. Our approach relies
on the intrinsic energy structure and on an adequate nonlinear and curiously
more regular approximate problem. The delicate passing to the limit is done in
adequate functional spaces which lead to only weak solutions.
