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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 ofdimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$,$\lambda\geq0$, $\mu\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Usingsemi-Riemannian isoparametric functions on $M$, we reduce the PDE into ageneralized Emden-Fowler ODE of the form \[ w''+q(r)w'+\lambda w = \mu\vertw\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$or $[0,\pi]$, $q(r)$ blows-up at $0$ and $w$ is subject to the natural initialconditions $w'(0)=0$ in the first case and $w'(0)=w'(\pi)=0$ in the second. Weprove the existence of blowing-up and globally defined solutions to thisproblem, both positive and sign-changing, inducing solutions to thesemi-Riemannian Yamabe type problem with the same qualitative properties, withlevel and critical sets described in terms of semi-Riemannian isoparametrichypersurfaces and focal varieties. In particular, we prove the existence ofsign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in thepseudosphere having a prescribed number of nodal domains.

An ODE reduction method for the semi-Riemannian Yamabe problem on space forms