Multiplicity of 2-nodal solutions the Yamabe equation

TL;DR

This paper proves the existence of multiple sign-changing solutions to a Yamabe-type equation on closed Riemannian manifolds using gradient flow and critical point theory, including results on product manifolds with small perturbations.

Contribution

It introduces new multiplicity results for 2-nodal solutions of subcritical Yamabe equations on general manifolds and their products, employing advanced variational methods.

Findings

Multiple 2-nodal solutions established for Yamabe equations.

Results extend to product manifolds with small perturbations.

Uses gradient flow and Sign-Changing Critical Point Theory.

Abstract

Given any closed Riemannian manifold $(M, g)$, we use the gradient flow method and Sign-Changing Critical Point Theory to prove multiplicity results for 2-nodal solutions of a subcritical Yamabe type equation on $(M, g)$. If $(N, h)$ is a closed Riemannian manifold of constant positive scalar curvature our result gives multiplicity results for the type Yamabe equation on the Riemannian product $(M x N, g + \epsilon h)$, for $\epsilon > 0$ small.