On a Conjecture of Bahri-Xu

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 $p\le 3$. 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 $m\ge 2$, we manage to reduce them to the basic case $m=1$ and thus prove them as well.

Accepted by Acta Math. Sin. (Engl. Ser.)