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In this paper, we deal with the following class of systems of nonlinear singular elliptic equations, involving critical Sobolev exponent, and coupled by a nonlinear termwhereΩ（?） RN（N≥3） is a smooth bounded domain, 0∈Ω, 0≤μ1,μ2＜（?） : = （?） is the critical Sobolev exponent,λ> 0,α＞1,β＞1.α+β≤2*. We are interested in the case that the existence of least energy solutions and asymptoticbehavior of solutions for （1.1）. Since the lack of compactness of the embedding H01（Ω） （?） L2*（Ω） and H01（Ω） （?） L2(Ω,|x|-2dx) could lead to non-convergence of （PS） sequences, we cannot use the standard variational argument directly. To overcomethis difficulty, we perform a careful analysis of the behavior of minimizing sequences with the aid of concentration-compactness principle due to P.L.Lions, which allows us to get the local （PS）c condition. Using this fact, we get the existenceof least energy solution. Moreover, we prove asymptotic behavior of solutions for （1.1） by the Moser iteration method.
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