# Improved Beckner's inequality for axially symmetric functions on $S_{4}$

### Changfeng Gui

University of Macau, China; The University of Texas at San Antonio, USA### Yeyao Hu

Central South University, Changsha, Hunan, China### Weihong Xie

Central South University, Changsha, Hunan, China

## Abstract

We show that axially symmetric solutions on $S_{4}$ to a constant $Q$-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter $α$ in front of the Paneitz operator belongs to the interval $[1800473+209329 ≈0.517,1)$. This is in contrast to the case $α=1$, where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on $S_{2}$. As a consequence, we prove an improved Beckner's inequality on $S_{4}$ for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when $α=1/5$ by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for $α∈(1/5,1/2)$ via a bifurcation method.

## Cite this article

Changfeng Gui, Yeyao Hu, Weihong Xie, Improved Beckner's inequality for axially symmetric functions on $S_{4}$. Rev. Mat. Iberoam. 40 (2024), no. 1, pp. 355–388

DOI 10.4171/RMI/1445