= Solution
For large positive dimensionless surface potential $u_0=e\Phi_0/(k_BT)\gtrsim1$, use the nonlinear <Boltzmann distribution> rather than their linear expansion:
$$
n_{\rm co}=n_0e^{-u(x)},\qquad n_{\rm counter}=n_0e^{u(x)}.
$$
The <counterions> become strongly enriched near the wall and <co-ions> strongly depleted. A planar mean-field solution is
$$
u(x)=4\operatorname{artanh}\!\left[\tanh(u_0/4)e^{-x/\lambda_D}\right].
$$
It approaches the linear screening form far from the wall. \b[Nonlinearity preserves positive densities and produces a much stronger near-wall asymmetry]; an extrapolated linear <co-ion> profile could incorrectly become negative. The total local ion density is $2n_0\cosh u$, no longer constant.
The original figure compares weak- and strong-potential profiles. At very large local concentrations, finite ion size, specific adsorption and ion correlations limit this ideal point-ion mean-field model; it does not predict unlimited physically realizable <counterion> packing.
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