An upper bound for the least energy of a nodal solution to the Yamabe equation on the sphere

Abstract

For each $n\ge 3$ we establish the existence of a nodal solution $u$ to the Yamabe problem on the round sphere $({\mathbb{S}}^n,g)$ which satisfies $${\int }_{{\mathbb{S}}^n}|u{|}^{2^{\ast }}d{V}_g<2m_n\mathrm{vol}({\mathbb{S}}^n),$$ where $m_3=9,$ $m_4=7,$ $m_5=m_6=6$, and $m_n=5\text{if }n\ge 7.$