Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

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.