Solutions of the Yamabe Equation By Lyapunov-Schmidt Reduction
Given any closed Riemannian manifold $(M,g)$ we use the Lyapunov-Schmidt finite-dimensional reduction method and the classical Morse and Lusternick-Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on $(M,g)$. If $(N,h)$ is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product $(M\times N , g + \ve^2 h )$, for $\ve >0$ small. For example, if $M$ is a closed Riemann surface of genus ${\bf g}$ and $(N,h) = (S^2 , g_0)$ is the round 2-sphere, we prove that for $\ve >0$ small enough and a generic metric $g$ on $M$, the Yamabe equation on $(M\times S^2 , g + \ve^2 g_0 )$ has at least $2 + 2 {\bf g}$ solutions.