Summary: We classify all positive solutions for the following integral system:
, , and . Here , , are real-valued functions of homogeneous degree and are monotone nondecreasing with respect to all the independent variables . In the special case and , we show that the above system is equivalent to the following elliptic PDE system:
This system is closely related to the stationary Schrödinger system with critical exponents for Bose-Einstein condensate.