Fractional nonhomogeneous system with Hardy-Littlewood-Sobolev critical nonlinearity

In this paper, we consider the following fractional elliptic system of Choquard type in RN {(−Δ)su=(∫RN|v(y)|2μ∗|x−y|μdy)|u|2μ∗−2u+f1(x)inRN,(−Δ)sv=(∫RN|u(y)|2μ∗|x−y|μdy)|v|2μ∗−2v+f2(x)inRN,u,v∈Ds(RN), where s∈(0,1), N>2s, 0<μ<min{N,4s}, 2μ∗=2N−μN−2s is the upper critical exponent in the Hardy-Littlewood-Sobolev inequality and f1,f2 are nonnegative functionals in the dual space of Ds(RN). When f1=f2=0, we establish the uniqueness of the solution to the above problem. On the other hand, when f1 and f2 are nonnegative functionals with Ker(f1)=Ker(f2), the multiplicity of solutions to the above problem is also shown. Moreover, we obtain the global compactness result by using (PS) decomposition.