Multiplicity of nodal solutions to the Yamabe problem

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 $$-di{v}_g(a\mathrm{\nabla }u)+bu=c\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}uonM$$ where $a,b,c\in C^{\infty}(M)$,$a$and$c$arepositive,$-div_{g}(a\nabla)+b$ 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 $${\mathrm{\Delta }}_{g}u+\frac{m-2}{4(m-1}{R}_{g}u=\kappa u^{2^{\ast }-2}onM$$ admits a prescribed number of nodal solutions.