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

TL;DR

This paper establishes the uniqueness of non-negative solutions for high-order Lane-Emden equations on the sphere, using integral equation methods and symmetry arguments, and identifies sharp constants for related Sobolev inequalities.

Contribution

It introduces a novel approach combining Möbius transforms and integral methods to prove uniqueness and classify extremals for high-order Sobolev inequalities on spheres.

Findings

Proved uniqueness of non-negative solutions for high-order Lane-Emden equations on n.

Identified best constants and classified extremals for sharp subcritical high-order Sobolev inequalities.

Extended the understanding of high-order elliptic equations on spheres beyond existing literature.

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 $\mathbb{S}^n$.