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

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 \begin{document}$ \mathbb{S}^n $\end{document} . Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in \begin{document}$ \mathbb{S}^n $\end{document} , we first employ the Mobius transform between \begin{document}$ \mathbb{S}^n $\end{document} and \begin{document}$ \mathbb{R}^n $\end{document} , poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on \begin{document}$ \mathbb{S}^n $\end{document} is equivalent to some integral equation in \begin{document}$ \mathbb{R}^n $\end{document} . Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on \begin{document}$ \mathbb{S}^n $\end{document} . As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on \begin{document}$ \mathbb{S}^n $\end{document} . Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on \begin{document}$ \mathbb{S}^n $\end{document} .