Multiplicity of nodal solutions to the Yamabe problem

Abstract: Given a compact Riemannian manifold $(M,g)$ without boundary of dimension $m\geq 3$ and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$-div_{g}(a\nabla u)+bu=c|u|^{{2^{{\ast}-2}u\quad}} on\ M$$ where $a,b,c\in C<sup>{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ is coercive, and $2<sup>{\ast}=\frac{2m}{m-2}$ 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 $$\Delta_{g}u+\frac{m-2}{4(m-1} R_{g}u=\kappa u^{{2^{{\ast}-2}\quad}} on\ M$$ admits a prescribed number of nodal solutions.