Electronic Journal of Differential Equations, Vol. 2004(2004), No. 138, pp. 1-8. Title: Existence results for elliptic systems involving critical Sobolev exponents Authors: Mohammed Bouchekif (Univ. of Tlemcen, Algeria) Yasmina Nasri (Univ. of Tlemcen, Algeria) Abstract: In this paper, we study the existence and nonexistence of positive solutions of an elliptic system involving critical Sobolev exponent perturbed by a weakly coupled term. Submitted July 6, 2004. Published November 25, 2004. Math Subject Classifications: 35J20, 35J50, 35J60. Key Words: Elliptic system; critical Sobolev exponent; variational method; moutain pass theorem