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

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 the interval $[\frac{473+\sqrt{209329}}{1800}\approx 0.517,1)$. This is in contrast to the case $\alpha =1$, where there exists a family of solutions, 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 $\alpha =1/5$ by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for $\alpha \in (1/5,1/2)$ via a bifurcation method.