On a Conjecture of Bahri-Xu

TL;DR

This paper proves a conjecture related to a universal inequality in the Yamabe problem for multiple points in Euclidean space, simplifying and extending previous verifications to higher dimensions and point counts.

Contribution

It simplifies the Bahri-Xu conjecture with necessary and sufficient conditions and proves it for all cases when the dimension is 1, and reduces higher-dimensional cases to this basic case.

Findings

Proved the conjecture for m=1 with arbitrary p.

Reduced cases p=4,5, m≥2 to the basic case m=1.

Extended verification of the conjecture beyond p≤3.

Abstract

In order to study the Yamabe changing-sign problem, Bahri and Xu proposed a conjecture which is a universal inequality for $p$ points in $\mathbb R^m$. They have verified the conjecture for $p\leq3$. In this paper, we first simplify this conjecture by giving two sufficient and necessary conditions inductively. Then we prove the conjecture for the basic case $m=1$ with arbitrary $p$. In addition, for the cases when $p=4,5$ and $m\geq2$, we manage to reduce them to the basic case $m=1$ and thus prove them as well.