Nodal solutions of Yamabe-type equations on positive Ricci curvature manifolds

We consider a closed cohomogeneity one Riemannian manifold (M^n,g) of dimension n\geq 3. If the Ricci curvature of M is positive, we prove the existence of infinite nodal solutions for equations of the form -\Delta _g u + \lambda u = \lambda u^q with \lambda >0, q>1. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeneity one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.