On a conjecture of Bahri-Xu (Q2063325)

The article concerns a conjecture posed by \textit{A. Bahri} and \textit{Y. Xu} [Recent progress in conformal geometry. London: Imperial College Press (2007; Zbl 1128.53002)]. The conjecture is the existence of a universal positive constant \(c(p,m)\) such that the following inequality \[ I_1\geq c(p,m) I_2, \] hl holds, where \begin{align*} I_1&=\sum_{1\leq i \leq p}\left|\sum_{i \neq j}\frac{u_j}{|x_i-x_j|}\right|^2+2\sup_{1\leq i \leq p}\left|\sum_{i \neq j}\frac{x_i-x_j}{|x_i-x_j|^3}u_i u_j\right|,\\ I_2&=\sum_{\substack{i,j\\ i \neq j }}\frac{u_j^2}{|x_i-x_j|^2}, \end{align*} and \(x_1, \ldots, x_p\) are distinct vectors in \(\mathbb R^m\), for any \(p\)-tuple \((u_1, u_2, \ldots u_p)\). The inequality arises in the derivation of a Morse-type lemma at infinity [\textit{Y. Xu}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 5, 629--639 (2006; Zbl 1112.53028)] for the Yamabe problem with changing signs. The inequality holds trivially when \(p=2\) with \(c(p,m)=1\) and the \(p=3\) case was proven by Xu in [loc. cit.]. In this article, the authors prove in Theorem 1.4 that the conjecture is true for all \(p\geq 4\) in Theorem 1.4 subject to the conditions that equations (1.4) given by \[ \sum_{i \neq j}\frac{u_j}{|x_i-x_j|}=0 \] and equations (1.5) given by \[ u_i\sum_{i \neq j}\frac{x_i-x_j}{|x_i-x_j|^3}u_j=0 \] for \(1\leq i \leq p\) admit no non-zero solutions \((u_1, u_2, \ldots u_p)\). Section 2 is devoted to the proof of Theorem 1.4. The proof relies on an inductive argument on \(p\) followed by an argument by contradiction on the components \(u_i\). Difficulty from singularities arises when the distance between one or more pairs of vectors \(x_i\) and \(x_j\) goes to \(0\) or infinity. Lemmas 2.1 and 2.2 deal with these situations. Section 3 uses the Kelvin transform to simplify the statement of Theorem 1.4, restating it as Theorem 1.5, and the authors show that the conditions given by (1.4) and (1.5) are equivalent to (1.6) (see the paper). The conjecture is then proven in the cases where \(m=1\), \(p\) is arbitrary (Theorem 3.2) and where \(m\) is arbitrary, \(p=4,5\) (Theorem 3.3).