  We consider the conformal class of the Riemannian product $g_0 + g$, where
$g_0$ is the constant curvature metric on $S^m$ and $g$ is a metric of constant
scalar curvature on some closed manifold. We show that the number of metrics of
constant scalar curvature in the conformal class grows at least linearly with
respect to the square root of the scalar curvature of $g$. This is obtained by
studying radial solutions of the equation $\Delta u -\lambda u + \lambda u^p
=0$ on $S^m$, and the number of solutions in terms of $\lambda$.
