= Solution
Substitution into <Poisson's equation>, $\nabla^2\psi=-\rho/\varepsilon$, gives the <Poisson-Boltzmann equation>
$$
\boxed{\nabla^2\psi=\frac{2qn_0}{\varepsilon}\sinh\!\frac{q\psi}{k_BT}.}
$$
For a plane, $\nabla^2\psi=d^2\psi/ds^2$; for a sphere, it is $r^{-2}d(r^2\psi')/dr$. The bulk conditions are $\psi\to0$ and $n_\pm\to n_0$. At a planar wall with negligible internal field and normal pointing into the liquid, <Gauss's law> imposes $-\varepsilon\psi'(0)=\sigma_s$; alternatively, a prescribed surface potential gives $\psi(0)=\psi_0$.
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