Analysis of a Nonlocal Poisson-Boltzmann Equation

A nonlinear, nonlocal dielectric continuum model, called the nonlocal modified PoissonBoltzmann equation (NMPBE), has been proposed to reflect the spatial-frequency dependence of dielectric permittivity in the calculation of electrostatics of ionic-solvated biomolecules. However, its analysis is difficult due to its definition involving Dirac delta distributions for modeling point charges, exponential nonlinear terms for ionic concentrations, and convolution terms for dielectric corrections. In this paper, these difficulties are overcome through using a solution decomposition, a non-symmetric representation of the variational problem, and a transformation to a variational system without any convolution involved. NMPBE is then proved to be a well posed mathematical model. This analysis establishes a foundation for the numerical solution and application of NMPBE.