  On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type
equation $-\Delta_g u + \lambda u = \lambda |u|^{q-1}u$, where $\lambda \in
\mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric
function $f$ with focal submanifolds of positive dimension. If $k>0$ is the
minimum of the dimensions of the focal submanifolds of $f$, we let $q^*
=\frac{n-k+2}{n-k-2}$. We prove the existence of infinite $f$-invariant
sign-changing solutions to the equation when $1<q<q^*$.
