In this paper, we consider the existence of a non-trivial weak solution to a quasilinear elliptic system involving critical Hardy exponents. The main issue of the paper is to understand the behavior of these Palais-Smale sequences. Indeed, the principal difficulty here is that there is an asymptotic competition between the energy functional carried by the critical nonlinearities. Then by the variational method, we obtain the existence non-trivial week solution for the system
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