  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 \[ -\Delta_{g_0}u+\lambda u=\lambda\ | u\ | ^{p-1}u\qquad\text{ on
}\mathbb{S}^n \] with $\lambda>0$ and $1<p$, into a singular ODE in $[0,\pi]$
of the form $w'' + \frac{h(r)}{\sin r} w' + \frac{\lambda}{\ell^2}\ (|
w|^{p-1}w - w\ )=0$, 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.
