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

Abstract

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