*主持人：叶东 教授
*时间：2020年11月13日8：30-9：30
*地点：腾讯会议 ID：197 139 648
*讲座内容简介：
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 non constant solutions exist, known as the standard bubbles. The phenomenon resembles the Gaussian curvature equation on $ \mathbb{S}^2$ in connection to the Moser-Trudinger-Onofri inequality. 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.
This is a joint work with Yeyao Hu and Weihong Xie from the Central South University.
