Improved Beckner's inequality for axially symmetric functions on S4

• Changfeng Gui ^[2] ; Yeyao Hu ^[1] ; Weihong Xie ^[1]
  1. [1] Central South University

     Central South University

     China

     
  2. [2] University of Macau, China; The University of Texas at San Antonio, USA
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 1, 2024, págs. 355-388
• Idioma: inglés
• DOI: 10.4171/RMI/1445
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• Resumen

  • 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.