Second Eigenvalue of the Yamabe Operator and Applications

Abstract

Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 3$. In this paper, we give various properties of the eigenvalues of the Yamabe operator $L_g$. In particular, we show how the second eigenvalue of $L_g$ is related to the existence of nodal solutions of the equation $L_g u = \epsilon | u|^{N-2} u$, where $\epsilon = +1, 0,$ or -1.