Metrics of constant scalar curvatures conformal to a Riemannian product with a round sphere

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 $\lambda$.