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

Abstract

We consider the semi-Riemannian Yamabe type equations of the form \[ -\square u + λu = μ\vert u\vert^{p-1}u\quad\text{ on }M \] where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\ge 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $\lambda \ge 0$, $\mu \in \mathbb{R}\setminus \{0\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+λw = μ\vert w\vert^{p-1}w\quad\text{ on } I, \] 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 $w^{\prime }(0)=0$ in the first case and $w^{\prime }(0)=w^{\prime }(\pi )=0$ 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.