On nodal solutions of the Yamabe equation on products

Let (M,g)(M,g), (N,h)(N,h) be closed Riemannian manifolds of constant scalar curvature. We prove the existence of nodal solutions of the Yamabe equation on the Riemannian product which depend on only one of the factors. We do this by studying the second Yamabe invariant introduced by Ammann and Humbert. We work out the case when M=S1M=S1 explicitly showing the existence of an infinite number of solutions.