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

We consider Yamabe-type equations on the Riemannian product of constant curvature metrics on \mathbf {S}^n \times \mathbf {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.