  We consider the nonlinear Poisson-Boltzmann equation in the context of
electrostatic models for a biological macromolecule, embedded in a bounded
domain containing a solution of an arbitrary number of ionic species which is
not necessarily charge neutral. The resulting semilinear elliptic equation
combines several difficulties: exponential growth and lack of sign preservation
in the nonlinearity accounting for ion mobility, measure data arising from
point charges inside the molecule, and discontinuous permittivities across the
molecule boundary. Exploiting the modelling assumption that the point sources
and the nonlinearity are active on disjoint parts of the domain, one can use a
linear decomposition of the potential into regular and singular components. A
variational argument can be used for the regular part, but the unbounded
nonlinearity makes the corresponding functional not differentiable in Sobolev
spaces. By proving boundedness of minimizers, these are related to standard
$H^1$ weak formulations for the regular component and in the framework of
Boccardo and Gallou\"et for the full potential. Finally, a result of uniqueness
of this type of weak solutions for more general semilinear problems with
measure data validates the strategy, since the different decompositions and
test spaces considered must then lead to the same solution.
