Global bifurcation for a class of nonlinear ODEs

TL;DR

This paper surveys global bifurcation methods and applies them to find multiple positive periodic solutions for certain nonlinear ODEs, providing a bifurcation-based proof of a classical nonuniqueness result in conformal geometry.

Contribution

It introduces a bifurcation-theoretic approach to establish nonuniqueness of solutions in a geometric PDE context, extending previous classical results.

Findings

Multiple positive periodic solutions for specific nonlinear ODEs

A bifurcation-based proof of nonuniqueness in conformal metrics

Application to Yamabe problem and scalar curvature

Abstract

We briefly survey global bifurcation techniques, and illustrate their use by finding multiple positive periodic solutions to a class of second order quasilinear ODEs related to the Yamabe problem. As an application, we give a bifurcation-theoretic proof of a classical nonuniqueness result for conformal metrics with constant scalar curvature, that was independently discovered by O. Kobayashi and R. Schoen in the 1980s.