On the Asymptotic Solution of a Partial Differential Equation with an Exponential Nonlinearity

Abstract

The asymptotic behavior of a large norm (maximum) solution of the Dirichlet problem associated with the equation \[ - \Delta u = \lambda e^u \] for a bounded simply-connected domain in $\mathbb{R}^2 $ is investigated for the case of the positive parameter $\lambda $ tending to zero. By means of a conformal transformation function $f(z)$, the problem is transformed to one involving the unit disc. For a class of domains which are described by implicit conditions for $f(z)$, a first and higher asymptotic expressions are developed for the large norm solution characterized by a single maximum proportional to $\ln ({1 / \lambda })$. It is shown that for $\lambda $ sufficiently small, an exact solution can be generated by the modified Newton iteration scheme, if the asymptotic solution of appropriate order is used for the initial step.