Variational properties of the second eigenvalue of the conformal Laplacian

Let (Mn,g) be a closed Riemannian manifold of dimension n?3. Assume [g] is conformal class for which the Conformal Laplacian Lg has at least two negative eigenvalues. We show existence (generalized) metric that maximizes second eigenvalue over all metrics (the first maximized by Yamabe metric). also maximal defines either nodal solution equation, or harmonic map to sphere. Moreover, we construct examples each possibility.