Stationary solutions to the Poisson-Nernst-Planck equations with steric effects

Ion transport, the movement of ions across a cellular membrane, plays a crucial role in a wide variety of biological processes and can be described by the Poisson-Nernst-Planck equations with steric effects (PNP-steric equations). In this paper, we shall show that under homogeneous Neumann boundary conditions, the steady-state PNP-steric equations are equivalent to a system of differential algebraic equations (DAEs). Analyzing this system of DAEs inspires us to propose an assumption on coupling constants, the so-called \textbf{(H1)} which will be introduced in \cref{Sec:model}, such that if \textbf{(H1)} holds true, the steady-state PNP-steric equations admit a unique stationary $C^2$ solution. Moreover, we shall point out the occurrence of bifurcation when \textbf{(H1)} is violated, which may relate to the opening and closing of the ion channels. When \textbf{(H1)} fails, we also suggest a simple criterion to check whether the system of DAE equations admits unique monotone $C^2$ solutions; or unique monotone piecewise $C^2$ solutions with vertical tangents; or triple piecewise $C^2$ solutions. To the best of the authors' knowledge, this is the first time such DAE approach has been utilized to obtain a complete investigation for the steady-state PNP-steric equations of two counter-charged ion species