Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

Given an isoparametric function $ f $ on the $ n $-dimensional round sphere, we consider functions of the form $ u = w\circ f $ to reduce the semilinear elliptic problem

$ -\Delta_{g_0}u+\lambda u = \lambda\left\vert u\right\vert ^{p-1}u\qquad\text{ on }\mathbb{S}^n $

with $ \lambda>0 $ and $ 1<p $, into a singular ODE in $ [0,\pi] $ of the form $ w" + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\left(\vert w\vert^{p-1}w - w\right) = 0 $, where $ h $ is an strictly decreasing function having exactly one zero in this interval and $ \ell $ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a sequence of sign-changing solutions to the first problem which are constant on the isoparametric hypersurfaces associated to $ f $ and blowing-up at one or two of the focal submanifolds generating the isoparametric family. Our methods apply also when $ p>\frac{n+2}{n-2} $, i.e., in the supercritical case. Moreover, using a reduction via harmonic morphisms, we prove existence and multiplicity of sign-changing solutions to the Yamabe problem on the complex and quaternionic space, having a finite disjoint union of isoparametric hipersurfaces as regular level sets.