Reverse Stein-Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein-Weiss inequality on the upper half space: for any nonnegative functions f 0 satisfying Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler-Lagrange equations of the reverse Stein-Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity.