Asymptotic analysis of the perturbed Poisson-Boltzmann equation on unbounded domains

摘要

We study the existence, uniqueness and asymptotic expansions to perturbed Poisson-Boltzmann equations on an unbounded domain in R~2 or R~3. First, a shooting method is applied to prove the existence and uniqueness of the exact solution. For the approximation to the regularly perturbed Poisson-Boltzmann equation, the solution via the classical method fails. We develop a novel approximate solution in terms of generalized asymptotic expansions. For the singularly perturbed problem, we show that a formula of asymptotic expansions with a boundary layer near the left end point provides a valid approximation. All our results are proved rigorously.