Qualitative Properties of Steady-State Poisson--Nernst--Planck Systems: Mathematical Study

We examine qualitative properties of solutions of self-consistent Poisson--Nernst--Planck systems, including uniqueness. In the case of vanishing permanent charge, the predominant case studied, our results unveil a rich structure inherent in these systems, one that is determined by the boundary conditions and the signs of the oppositely charged carrier fluxes. A particularly significant special case, that of simple boundary conditions, is shown to lead to uniqueness and to a complete characterization. This case underlies the more complicated cases studied later. A contraction mapping principle is included for completeness and allows for an arbitrary permanent charge distribution.