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

We study the existence, uniqueness and asymptotic expansions to perturbed Poisson–Boltzmann equations on an unbounded domain in R² or R³. 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.