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

TL;DR

This paper proves the existence of nodal solutions to the Yamabe problem on spheres with energy below a certain upper bound, providing explicit bounds for different dimensions.

Contribution

It establishes new upper bounds for the least energy of nodal solutions to the Yamabe equation on spheres for various dimensions.

Findings

Existence of nodal solutions with energy below specified bounds

Explicit bounds for dimensions 3 to 7 and higher

Advancement in understanding the energy landscape of Yamabe solutions

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 $m_n= 5\ \text{if }n\geq 7.$