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^β}{|x-y|^{n-α}}f(y)g(x) dydx\geq C_{n,α,β,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 $g\in L^{q^{\prime }}({\mathbb{R}}_{+}^n)$, where $n\ge 2$, $p,q^{\prime }\in (0,1)$, $\alpha >n$, $0\le \beta <\frac{\alpha -n}{n-1}$, $p>\frac{n-1}{\alpha -1-(n-1)\beta }$ such that $\frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime }}-\frac{\alpha +\beta -1}{n}=1$. 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 $\beta \ge 0$.