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

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 $[\frac{473 + \sqrt{209329}}{1800}\approx0.517, 1)$\mathrm{.}Thisisincontrasttothecase$\alpha=1$, 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 $\alpha =\frac{1}{5}$ by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for $\alpha \in (\frac{1}{5},\frac{1}{2})$ via a bifurcation method.