A note on sign-changing solutions to supercritical Yamabe-type equations

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 $1<q<q^*$.