Reversed Hardy-Littlewood-Sobolev inequalities with vertical weights on the upper half space

Abstract

In this paper, we obtain the reversed Hardy-Littlewood-Sobolev inequality with vertical weights on the upper half space and discuss the extremal functions. We show that the sharp constants in this inequality are attained by introducing a renormalization method. The classification of corresponding extremal functions is discussed via the method of moving spheres. Moreover, we prove the sufficient and necessary conditions of existence for positive solutions to the Euler-Lagrange equations by using Pohozaev identities in weak sense. This renormalization method is rearrangement free, which can be also applied to prove the existence of extremal functions for sharp (reversed) Hardy-Littlewood-Sobolev inequality with extended kernels and other similar inequalities.