  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.
