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

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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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  • 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 $.

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


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