The paper completes earlier results of the same authors [Zastosow. Mat. 21, No. 2, 265-272 (1991; Zbl 0756.35029)] showing the uniqueness of radially symmetric solutions of the Poisson-Boltzmann equation. The main result of the paper is the following Theorem: If \(f\) is Lipschitz continuous and positive in \(\mathbb{R}\), then for any positive \(\sigma\) the problem \[ -{d\over {dr}} \left( r^ 2 {{du} \over {dr}}\right)= \sigma\mu(u)r^ 2 f(u), \qquad \mu(u)= \left( \int^ 1_ a s^ 2 f[u(s)]ds\right)^{-1}, \tag{1} \]
\[ u(a)=0, \qquad {{du} \over {dr}} \Biggl|_{r=1}=-\sigma, \tag{2} \] has a unique solution.