Existence of Strong Solution for the Complexified Non-linear Poisson Boltzmann Equation

We prove the existence and uniqueness of the complexified Nonlinear Poisson-Boltzmann Equation (nPBE) in a bounded domain in $\mathbb{R}^3$. The nPBE is a model equation in nonlinear electrostatics. The standard convex optimization argument to the complexified nPBE no longer applies, but instead, a contraction mapping argument is developed. Furthermore, we show that uniqueness can be lost if the hypotheses given are not satisfied. The complixified nPBE is highly relevant to regularity analysis of the solution of the real nPBE with respect to the dielectric (diffusion) and Debye-H\"uckel coefficients. This approach is also well-suited to investigate the existence and uniqueness problem for a wide class of semi-linear elliptic Partial Differential Equations (PDEs).