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

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.