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

$$\begin{aligned} -\text {div}_{g}(a\nabla u)+bu=c|u|^{2^{*}-2}u\quad \text { on }M, \end{aligned}$$

where \(a,b,c\in \mathcal {C}^{\infty }(M), a\) and c are positive, ? div\(_{g}(a\nabla )+b\) is coercive, and \(2^{*}=\frac{2m}{m-2}\) 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

$$\begin{aligned} \Delta _{g}u+\frac{m-2}{4(m-1)}R_{g}u=\kappa u^{2^{*}-2}\quad \text { on }M \end{aligned}$$

admits a prescribed number of nodal solutions.