Resumen de Improved Beckner's inequality for axially symmetric functions on S4

Changfeng Gui, Yeyao Hu, Weihong Xie

• We show that axially symmetric solutions on S4 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 the interval [1800473+209329≈0.517,1). This is in contrast to the case α=1, where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on S2. As a consequence, we prove an improved Beckner's inequality on S4 for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when α=1/5 by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for α∈(1/5,1/2) via a bifurcation method.