  When ions are crowded, the effect of steric repulsion between ions (which can
produce oscillations in charge density profiles) becomes significant and the
conventional Poisson-Boltzmann (PB) equation should be modified. Several
modified PB equations were developed but the associated total ionic charge
density has no oscillation. This motivates us to derive a general model of PB
equations called the PB-steric equations with a parameter $\Lambda$, which not
only include the conventional and modified PB equations but also have
oscillatory total ionic charge density under different assumptions of steric
effects and chemical potentials. As $\Lambda=0$, the PB-steric equation becomes
the conventional PB equation, but as $\Lambda>0$, the concentrations of ions
and solvent molecules are determined by the Lambert type functions. To approach
the modified PB equations, we study the asymptotic limit of PB-steric equations
with the Robin boundary condition as $\Lambda$ goes to infinity. Our
theoretical results show that the PB-steric equations (for
$0\leq\Lambda\leq\infty$) may include the conventional and modified PB
equations. On the other hand, we use the PB-steric equations to find
oscillatory total ionic charge density which cannot be obtained in the
conventional and modified PB equations.
