Multiplicity of 2-nodal solutions of the Yamabe equation

Abstract

Given a closed Riemannian manifold $(M,g)$, we use the gradient flow method and Sign-Changing Critical Point Theory to prove multiplicity results for $2$-nodal solutions of a subcritical non-linear equation on $(M,g)$, see (1.1) below. If $(N,h)$ is a closed Riemannian manifold of constant positive scalar curvature our result gives multiplicity results for the Yamabe-type equation on the Riemannian product $(M\times N , g + \varepsilon h )$, for $\varepsilon > 0$ small.