Second Eigenvalue of the Yamabe Operator and Applications

TL;DR

This paper investigates the second eigenvalue of the Yamabe operator on compact Riemannian manifolds and explores its implications for the existence of nodal solutions to related nonlinear equations.

Contribution

It establishes new relationships between the second eigenvalue of the Yamabe operator and the existence of nodal solutions to specific nonlinear equations on manifolds.

Findings

Second eigenvalue linked to nodal solutions existence

Properties of Yamabe operator eigenvalues analyzed

Conditions for solutions based on eigenvalues derived

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.