2021/09/27 by Gui, Changfeng, Hu, Yeyao, Xie, Weihong
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2109.13390
We show that axially symmetric solutions on \mathbbS4 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 [(473 + √(209329))/(1800)≈0.517, 1). This is in contrast to the case α=1, where a family of solutions exist, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on \mathbbS2. As a consequence, we prove an improved Beckner's inequality on \mathbbS4 for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when α=\frac15 by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for α∈ (\frac15, \frac12) via a bifurcation method.