Solutions of the Yamabe Equation by Lyapunov–Schmidt Reduction

Abstract

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 \((M\times N , g + \varepsilon ^2 h )\), for \(\varepsilon >0\) small. For example, if M is a closed Riemann surface of genus \(\mathbf{g}\) and \((N,h) = (S^2 , g_0)\) is the round 2-sphere, we prove that for \(\varepsilon >0\) small enough and a generic metric g on M, the Yamabe equation on \((M\times S^2 , g + \varepsilon ^2 g_0 )\) has at least \(2 + 2 \mathbf{g}\) solutions.