A note on solutions of Yamabe-type equations on products of spheres

TL;DR

This paper investigates Yamabe-type equations on products of spheres, establishing the existence and multiplicity of solutions, including non-trivial nodal solutions invariant under a specific symmetry group.

Contribution

It provides new multiplicity results for solutions of Yamabe equations on product spheres, especially demonstrating the existence of non-trivial nodal solutions.

Findings

Existence of multiple positive solutions.

Existence of nodal solutions depending on both factors.

Solutions invariant under cohomogeneity one group action.

Abstract

We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on $\textbf{S}^n \times\textbf{ S}^n$, and study solutions which are invariant by the cohomogeneity one diagonal action of $O(n+1)$. We obtain multiplicity results for both positive and nodal solutions. In particular we prove the existence of nodal solutions of the Yamabe equation on these products which depend non-trivially on both factors