Improved Beckner's inequality for axially symmetric functions on S^4
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题目: Improved Beckner's inequality for axially symmetric functions on S^4
摘要: We show that axially symmetric solutions on S^4 to a constant Q-curvature type equation must be constant, provided that the parameter α in front of the Paneitz operator belongs to [0.517,1). This is in contrast to the case α=1, where a family of non constant solutions exist, known as the standard bubbles. The phenomenon resembles the Gaussian curvature equation on S^2 in connection to the Moser-Trudinger-Onofri inequality. As a consequence, we prove an improved Beckner's inequality on S^4 for axially symmetric functions with their centers of mass at the origin. The four dimensional counterpart of first Szego limit Theorem is also established as a byproduct. Furthermore, we show uniqueness of axially symmetric solutions when α=1/5 by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for α∈(1/5,1/2) via a bifurcation method. This is a joint work with Professor Changfeng Gui (UTSA) and Weihong Xie (CSU).