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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula>. Since the classical moving-plane method based on the Kelvin transform and maximum principle fails in dealing with the high-order elliptic equations in <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula>, we first employ the Mobius transform between <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, poly-harmonic average and iteration arguments to show that the higher-order Lane-Emden equation on <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula> is equivalent to some integral equation in <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. Then we apply the method of moving plane in integral forms and the symmetry of sphere to obtain the uniqueness of nonnegative solutions to the higher-order Lane-Emden equations with subcritical polynomial growth on <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula>. As an application, we also identify the best constants and classify the extremals of the sharp subcritical high-order Sobolev inequalities involving the GJMS operators on <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula>. Our results do not seem to be in the literature even for the Lane-Emden equation and sharp subcritical Sobolev inequalities for first order derivatives on <inline-formula><tex-math id="M10">\begin{document}$ \mathbb{S}^n $\end{document}</tex-math></inline-formula>.</p>