Reversed Hardy-Littlewood-Sobolev Type Inequality on $${\mathbb {R}}^{n-m}\times {\mathbb {R}}^{n}$$

Abstract

In this paper, we are going to establish a reversed Hardy–Littlewood–Sobolev inequality on different dimensional space and prove the existence of extremal functions for the best constant. Furthermore, we investigate the regularity of extremal functions and provide a necessary conditions for the existence of solutions of the integral systems. Finally, we classify the extremal functions.