Classification of positive solutions for nonlinear differential and integral systems with critical exponents

We classify all positive solutions for the following integral system: {ui(x) = ∫Rn1|x−y|n−afi(u(y)dy, x∈Rn, i = 1,…,m,0<α 3 and α = 2, we show that the above system is equivalent to the following elliptixc PDE system: {−Δui(x) = fi(u(x)), x ∈ Rn, i = 1,…, m,and u(x) = (u1(x),u2(x),…,um(x)).This system is closely related to the stationary Schrodinger system with critical exponents for Bose-Einstein condensate.