Multiplicity of nodal solutions to the Yamabe problem

Abstract

Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\ge 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-di{v}_g(a\nabla u)+bu=c|u{|}^{2^{\ast }-2}uonM$$ where $a,b,c\in C^{\infty }(M)$, $a$ and $c$ are positive, $-di{v}_g(a\nabla )+b$ is coercive, and $2^{\ast }=\frac{2m}{m-2}$ is the critical Sobolev exponent. In particular, if $R_g$ denotes the scalar curvature of $(M,g)$, we give conditions which guarantee that the Yamabe problem $${\Delta }_{g}u+\frac{m-2}{4(m-1}{R}_{g}u=\kappa u^{2^{\ast }-2}onM$$ admits a prescribed number of nodal solutions.