EXISTENCE OF POSITIVE SOLUTIONS FOR INTEGRAL SYSTEMS OF THE WEIGHTED HARDY-LITTLEWOOD-SOBOLEV TYPE

This paper is concerned with the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system. Such a system is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality. The Serrin-type condition is critical for existence of positive solutions in L-lo(c)infinity (R-n \ {0}). When the Serrin-type condition does not hold, we prove the nonexistence by an iteration process. In addition, we find three pairs of radial solutions when the Serrin-type condition holds. One is singular, and the other two are integrable in R-n and decaying fast and slowly respectively.