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

Abstract

We consider a closed cohomogeneity one Riemannian manifold ( M n , g ) (M^n,g) of dimension n ≥ 3 n\geq 3 . If the Ricci curvature of M M is positive, we prove the existence of infinite nodal solutions for equations of the form − Δ g u + λ u = λ u q -\Delta _g u + \lambda u = \lambda u^q with λ > 0 \lambda >0 , q > 1 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.