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: integral(R+n) integral(partial derivative R+n) vertical bar x vertical bar(alpha)vertical bar x-y vertical bar(lambda)f(x)g(y)vertical bar y vertical bar(beta) dy dx >= C-n,C-alpha,C-beta,C-p,C-q'parallel to f parallel to(Lq'(R+n))parallel to g parallel to(L)p((partial derivative R+n)) for any nonnegative functions f is an element of L-q' (R-+(n)), g is an element of L-p(partial derivative R-+(n)), and p, q' is an element of (0,1), beta < 1-n/p' or alpha < -n/q, lambda > 0 satisfying n-1/n 1/p + 1/q' - alpha + beta + lambda - 1/n = 2. 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. Finally, in view of the stereographic projection, we give a spherical form of the Stein-Weiss inequality and reverse Stein-Weiss inequality on the upper half space R-+(n).