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

TL;DR

This paper develops an ODE reduction method for semi-Riemannian Yamabe equations on space forms, establishing existence of solutions with prescribed properties using isoparametric functions, and analyzing their qualitative behavior.

Contribution

It introduces a novel reduction of the semi-Riemannian Yamabe PDE to a generalized Emden-Fowler ODE using isoparametric functions, enabling detailed analysis of solutions.

Findings

Existence of blowing-up solutions with prescribed nodal domains.

Construction of sign-changing solutions with specific qualitative properties.

Description of solutions' level and critical sets via isoparametric hypersurfaces.

Abstract

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.