Systems of elliptic equations involving multiple critical nonlinearities and different Hardy-type terms in RN

In this paper, we study nonlinear systems of elliptic equations involving different Hardy-type terms and multiple critical nonlinearities. By very technical variational methods, local compactness of Palais-Smale sequences is established and values of the best Sobolev-type constants are compared. Finally, the existence of minimizers to related Rayleigh quotient is verified and the existence of positive solutions to elliptic systems is thus proved.