  We consider the semi-Riemannian Yamabe type equations of the form
  \[
  -\square u + \lambda u = \mu \vert u\vert^{p-1}u\quad\text{ on }M
  \] where $M$ is either the semi-Euclidean space or the pseudosphere of
dimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$,
$\lambda\geq0$, $\mu\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Using
semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a
generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+\lambda w = \mu\vert
w\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$
or $[0,\pi]$, $q(r)$ blows-up at $0$ and $w$ is subject to the natural initial
conditions $w'(0)=0$ in the first case and $w'(0)=w'(\pi)=0$ in the second. We
prove the existence of blowing-up and globally defined solutions to this
problem, both positive and sign-changing, inducing solutions to the
semi-Riemannian Yamabe type problem with the same qualitative properties, with
level and critical sets described in terms of semi-Riemannian isoparametric
hypersurfaces and focal varieties. In particular, we prove the existence of
sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the
pseudosphere having a prescribed number of nodal domains.
