REVERSE STEIN-WEISS INEQUALITIES AND EXISTENCE OF THEIR EXTREMAL FUNCTIONS

In this paper, we establish the following reverse Stein-Weiss inequality, namely the reversed weighted Hardy-Littlewood-Sobolev inequality, in R-n: integral(Rn) integral(Rn) vertical bar x vertical bar(alpha)vertical bar x-y vertical bar(lambda)f(x)g(y)vertical bar y vertical bar(beta) dxdy >= C-n,C-alpha,C-beta,C-p,C-q'parallel to f parallel to(Lq')parallel to g parallel to L-p for any nonnegative functions f is an element of L-q' (R-n), g is an element of L-p(R-n), and p, q' is an element of(0, 1), alpha, beta, lambda > 0 such that 1/p + 1/q' - alpha+beta+lambda/n = 2. We derive the existence of extremal functions for the above inequality. Moreover, some asymptotic behaviors are obtained for the corresponding Euler-Lagrange system. For an analogous weighted system, we prove necessary conditions of existence for any positive solutions by using the Pohozaev identity. Finally, we also obtain the corresponding Stein-Weiss and reverse Stein-Weiss inequalities on the n-dimensional sphere S-n by using the stereographic projections. Our proof of the reverse Stein-Weiss inequalities relies on techniques in harmonic analysis and differs from those used in the proof of the reverse (non-weighted) Hardy-Littlewood-Sobolev inequalities.