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

We consider a closed cohomogeneity one Riemannian manifold $(M^n,g)$ of dimension $n\ge 3$. 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 cohomogeniety 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.