Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$

TL;DR

This paper establishes conditions under which axially symmetric solutions to a fourth-order geometric PDE on the 4-sphere are constant, leading to improved inequalities and insights into solution uniqueness and existence.

Contribution

It provides new uniqueness results, improved inequalities, and existence proofs for axially symmetric solutions to a Q-curvature type equation on -sphere, depending on the parameter .

Findings

Axially symmetric solutions are constant for  in [0.517, 1).

An improved Beckner's inequality for axially symmetric functions on .

Existence of non-constant solutions for  in (/5, 1/2).

Abstract

We show that axially symmetric solutions on $\mathbb{S}^4$ to a constant $Q$-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter $\alpha$ in front of the Paneitz operator belongs to $[\frac{473 + \sqrt{209329}}{1800}\approx0.517, 1)$. This is in contrast to the case $\alpha=1$, where a family of solutions exist, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on $ \mathbb{S}^2$. As a consequence, we prove an improved Beckner's inequality on $\mathbb{S}^4$ for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when $\alpha=\frac15$ by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for $\alpha \in (\frac15, \frac12)$ via a bifurcation method.