Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space

In this paper we mainly classify the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space \begin{document}$\mathbb{R}_+^{n}$\end{document} , and also present some remarks on the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the whole space \begin{document}$\mathbb{R}^{n}$\end{document} . Our main techniques are Kelvin transformation and the method of moving spheres in integral forms.