Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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).

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The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom