Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $

Lu Chen; Guozhen Lu; Yansheng Shen

doi:10.3934/cpaa.2022073

Article Contents

2022, Volume 21, Issue 8: 2799-2817. Doi: 10.3934/cpaa.2022073

This issue Previous Article Global Carleman estimate and its applications for a sixth-order equation related to thin solid films Next Article Minimisers of Helfrich functional for surfaces of revolution

Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $

1.
School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China
2.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA
3.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

^* Corresponding author
^* Corresponding author

Received: November 30, 2021

Revised: January 31, 2022

Early access: April 2022

Published: August 2022

The first author was partly supported by NSFC (No.11901031). The second author was partly supported by a grant from the Simons Foundation

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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 $ \mathbb{S}^n $. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in $ \mathbb{S}^n $, we first employ the Mobius transform between $ \mathbb{S}^n $ and $ \mathbb{R}^n $, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on $ \mathbb{S}^n $ is equivalent to some integral equation in $ \mathbb{R}^n $. 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 $ \mathbb{S}^n $. 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 $ \mathbb{S}^n $. Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on $ \mathbb{S}^n $.

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Mathematics Subject Classification: 5J30, 46E35, 35B06, 35A02.

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