Multiplicity results for the Yamabe equation by Lusternik–Schnirelmann theory

Abstract

Let (M,g) be any closed Riemannianan manifold and (N,h) be a Riemannian manifold of constant positive scalar curvature. We prove that the Yamabe equation on the Riemannian product (M×N,g+δh) has at least Cat(M)+1 solutions for δ small enough, where Cat(M) denotes the Lusternik–Schnirelmann-category of M. The solutions obtained are functions of M and Cat(M) of them have energy arbitrarily close to the minimum.