  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).
