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 + h u = |u|^{q-1}u $, where $ q>1 $ and $ h $ is a positive $ C^1 $- function. 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^* $.