Matched asymptotics for large solutions to the Gelfand-Liouville problem in two-dimensional, doubly connected domains

TL;DR

This paper develops a formal asymptotic analysis and constructs approximate solutions for large solutions to the Gelfand-Liouville problem in doubly connected planar domains, advancing understanding of these nonlinear PDEs.

Contribution

It introduces a formal matched asymptotic method and rigorously constructs approximate solutions for the Gelfand-Liouville problem in complex geometries.

Findings

Successful asymptotic analysis for large solutions

Construction of accurate approximate solutions

Enhanced understanding of the problem's behavior in doubly connected domains

Abstract

In this paper we provide a formal matched asymptotic analysis for large solutions to the Gelfand-Liouville problem in planar, doubly connected domains in the plane. Using these, we rigorously construct a good approximate solution to the problem.