  We showed the existence of non-radial solutions of the equation $\Delta u
-\lambda u + \lambda u^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and
study the number of such solutions in terms of $\lambda$. We show that for any
isoparametric hypersurface $M\subset S^m$ there are solutions such that $M$ is
a regular level set (and the number of such solutions increases with
$\lambda$). We also show similar results for isoparametric hypersurfaces in
general Riemannian manifolds. These solutions give multiplicity results for
metrics of constant scalar curvature on conformal classes of Riemannian
products.
