  When ions are crowded, the effect of steric repulsion between ions becomes
significant and the conventional Poisson--Boltzmann (PB) equation (without
steric effect) should be modified. For this purpose, we study the asymptotic
limit of steady state Poisson--Nernst--Planck equations with steric effects
(PNP-steric equations). By the assumptions of steric effects, we transform
steady state PNP-steric equations into a PB equation with steric effects
(PB-steric equation) which has a parameter $\Lambda$ and positive constants
$\lambda_i$'s (depend on the radii of ions and solvent molecules). The
nonlinear term of PB-steric equation is mainly determined by a Lambert type
function which represents the concentration of solvent molecules. As
$\Lambda=0$, the PB-steric equation becomes the conventional PB equation but as
$\Lambda>0$, a large $\Lambda$ makes the steric repulsion (between ions and
solvent molecules) stronger. This motivates us to find the asymptotic limit of
PB-steric equation as $\Lambda$ goes to infinity. Under the Robin (or Neumann)
boundary condition, we prove theoretically and numerically that the PB-steric
equation has a unique solution $\phi_\Lambda$ which converges to solution
$\phi^*$ of a modified PB (mPB) equation as $\Lambda$ tends to infinity. Our
results show that the limiting equation of PB-steric equation (as $\Lambda$
goes to infinity) is a mPB equation which has the same form (up to scalar
multiples) as those mPB equations in
\cite{1942bikerman,1997borukhov,2007kilic,2009li,2009li2,2013li,2011lu}.
Therefore, the PB-steric equation can be regarded as a generalized model of mPB
equations.
