  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.
