Improved Beckner's inequality for axially symmetric functions on
Changfeng Gui
University of Macau, China; The University of Texas at San Antonio, USAYeyao Hu
Central South University, Changsha, Hunan, ChinaWeihong Xie
Central South University, Changsha, Hunan, China
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Abstract
We show that axially symmetric solutions on to a constant -curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter in front of the Paneitz operator belongs to the interval . This is in contrast to the case , where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on . As a consequence, we prove an improved Beckner's inequality on for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for via a bifurcation method.
Cite this article
Changfeng Gui, Yeyao Hu, Weihong Xie, Improved Beckner's inequality for axially symmetric functions on . Rev. Mat. Iberoam. 40 (2024), no. 1, pp. 355–388
DOI 10.4171/RMI/1445