Integral inequalities with an extended Poisson kernel and the existence of the extremals

Abstract In this article, we first apply the method of combining interpolation theorem and weak-type estimate developed in Chen et al. to derive Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using weighted Hardy inequality, further obtain Stein-Weiss For extremal problem corresponding presence double-weighted exponents not being necessarily nonnegative makes it impossible desired existence result through usual technique symmetrization rearrangement. We then adopt concentration compactness principle integral operator, which was used by authors overcome difficulty extremals. Finally, regularity positive solution for system related kernel is also considered article. Our avoids nonnegativity condition exponents, a common assumption dealing solutions systems literatures.