SHARP SUBCRITICAL SOBOLEV INEQUALITIES AND UNIQUENESS OF NONNEGATIVE SOLUTIONS TO HIGH-ORDER LANE-EMDEN EQUATIONS ON Sn

Lu Chen, Guozhen Lu, Yansheng Shen*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

In this paper, we are concerned with the uniqueness result for non-negative solutions of the higher-order Lane-Emden equations involving the GJMS operators on Sn. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in Sn, we first employ the Mobius transform between Sn and Rn, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on Sn is equivalent to some integral equation in Rn. Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on Sn. As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on Sn.

Original languageEnglish
Pages (from-to)2799-2817
Number of pages19
JournalCommunications on Pure and Applied Analysis
Volume21
Issue number8
DOIs
Publication statusPublished - Aug 2022

Keywords

  • Lane-Emden equations
  • Uniqueness of nonnegative solutions
  • extremal functions
  • moving-plane method
  • sharp constants

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