# Global bifurcation techniques for Yamabe type equations on Riemannian manifolds

## TL;DR

This paper develops global bifurcation methods to analyze positive solutions of Yamabe-type equations on Riemannian manifolds, establishing conditions for infinite solution multiplicity as parameters vary, especially in critical and supercritical cases.

## Contribution

It introduces new bifurcation techniques for Yamabe equations on manifolds with isoparametric functions, showing solution multiplicity results in supercritical regimes.

## Key findings

- Number of solutions tends to infinity as λ→∞ for certain q
- Multiplicity results for critical and supercritical equations
- Application to Yamabe problem on Riemannian manifolds

## Abstract

We consider a closed Riemannian manifold $(M^n ,g)$ of dimension $n\geq 3$ and study positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$, with $\lambda >0$, $q>1$. If $M$ supports a proper isoparametric function with focal varieties $M_1$, $M_2$ of dimension $d_1 \geq d_2 $ we show that for any $q<\frac{ n-d_2+2 }{n - d_2 -2}$ the number of positive solutions of the equation $-\Delta_g u + \lambda u = \lambda u^q$ tends to $\infty$ as $\lambda \rightarrow +\infty$. We apply this result to prove multiplicity results for positive solutions of critical and supercritical equations. In particular we prove multiplicity results for the Yamabe equation on Riemannian manifolds.