Sharp reversed Hardy-Littlewood-Sobolev inequality with extended kernel

Abstract: In this paper, we prove the following reversed Hardy-Littlewood-Sobolev inequality with extended kernel \begin{equation*} \int_{\mathbb{R}+^n}\int{\partial\mathbb{R}^n_+} \frac{x_n^\beta}{|x-y|^{n-\alpha}}f(y)g(x) dydx\geq C_{n,\alpha,\beta,p}|f|{L^{p}(\partial\mathbb{R}+^n)} |g|{L^{q'}(\mathbb{R}+^n)} \end{equation*} for any nonnegative functions $f\in L^{p}(\partial\mathbb{R}_+^n)$ and $g\in L^{q'}(\mathbb{R}_+^n)$, where $n\geq2$, $p,\ q'\in (0,1)$, $\alpha>n$, $0\leq\beta<\frac{\alpha-n}{n-1}$, $p>\frac{n-1}{\alpha-1-(n-1)\beta}$ such that $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}-\frac{\alpha+\beta-1}{n}=1$. We prove the existence of extremal functions for the above inequality. Moreover, in the conformal invariant case, we classify all the extremal functions and hence derive the best constant via a variant method of moving spheres, which can be carried out \emph{without lifting the regularity of Lebesgue measurable solutions}. Finally, we derive the sufficient and necessary conditions for existence of positive solutions to the Euler-Lagrange equations by using Pohozaev identities. Our results are inspired by Hang, Wang and Yan \cite{HWY}, Dou, Guo and Zhu \cite{DGZ} for $\alpha<n$ and $\beta=1$, and Gluck \cite{Gl} for $\alpha<n$ and $\beta\geq0$.