  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 $a,b,c\in
C^{\infty}(M)$, $a$ and $c$ are positive, $-div_{g}(a\nabla)+b$ 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 $$\Delta_{g}u+\frac{m-2}{4(m-1} R_{g}u=\kappa
u^{2^{\ast}-2}\quad on\ M$$ admits a prescribed number of nodal solutions.
