Local bifurcation diagrams and degenerate solutions of Yamabe-type equations

TL;DR

This paper investigates positive solutions of a Yamabe-type equation on the sphere, using bifurcation theory and isoparametric functions to analyze solution structure, degeneracy, and multiplicity of conformal metrics.

Contribution

It introduces a reduction to ODEs via isoparametric functions and characterizes bifurcation points, providing new insights into degenerate solutions and metric multiplicity.

Findings

Identification of transcritical bifurcation points.

Existence of degenerate solutions to the Yamabe-type equation.

Results on multiplicity of conformal constant scalar curvature metrics.

Abstract

We study positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$, with $\lambda >0$, $q>1$ on the round sphere $\mathbb{S}^n$ . We reduce the equation to an ordinary differential equation by considering isoparametric functions and apply bifurcation theory. We study when the corresponding bifurcation points are transcritical. We apply this result to show the existence of degenerate solutions to the equation and to study multiplicity results for conformal constant scalar curvature metrics.