Abstract
In this paper we use the maximum principle, simple comparision method and a priori estimates of rubsolutions for the scalar elliptic equations to obtain the asymptotic properties for a class of semi-linear elliptic systems involving Sobolev critical exponent in RN. Furthermore, we show that the positive solutions obtained by the Mountain pass lemma is a critical point of a set of natural constraints. Hence we establish a bifurcation result for this system.