On a local theory of asymptotic integration for nonlinear ordinary differential equations

TL;DR

This paper revisits and enhances an asymptotic integration theory for nonlinear ODEs, applying it to establish the existence of bounded positive solutions for certain semi-linear elliptic PDEs.

Contribution

It improves and generalizes classical asymptotic integration results and applies them to nonlinear PDEs using the subsolution-supersolution method.

Findings

Enhanced asymptotic integration framework for nonlinear ODEs

New existence results for bounded positive solutions of semi-linear elliptic PDEs

Broadened applicability of subsolution-supersolution approach

Abstract

By revisiting an asymptotic integration theory of nonlinear ordinary differential equations due to J.K. Hale and N. Onuchic [Contributions Differential Equations 2 (1963), 61--75], we improve and generalize several recent results in the literature. As an application, we study the existence of bounded positive solutions to a large class of semi-linear elliptic partial differential equations via the subsolution-supersolution approach.