Abstract: In this paper, we study the following perturbation problem with Sobolev critical exponent: \begin{equation}\label{eqs0.1} \left\{ \begin{array}{ll} -\Delta u=(1+\varepsilon K(x)){{u}^{{{2}^{*}}-1}}+\frac{\alpha }{{{2}^{*}}}{{u}^{\alpha -1}}{{v}^{\beta }}+\varepsilon h(x){{u}^{p}},\ \ &x\in \mathbb{R}^N,\\[2.5mm] -\Delta v=(1+\varepsilon Q(x)){{v}^{{{2}^{*}}-1}}+\frac{\beta }{{{2}^{*}}}{{u}^{\alpha }}{{v}^{\beta -1}}+\varepsilon l(x){{v}^{q}},\ \ &x\in \mathbb{R}^N,\\[2mm] u> 0,\,v> 0,\ \ &x\in \mathbb{R}^N, \end{array} \right. \end{equation} where $0 < p,\,q < 1$, $\alpha +\beta ={{2}^{*}}:=\frac{2N}{N-2}$, $\alpha,\,\beta\geq 2$ and $N=3, 4$. Using a perturbation argument and a finite dimensional reduction method, we get the existence of positive solutions to problem (0.1) and the asymptotic property of the solutions.

Key words: perturbation argument, finite dimensional reduction method, critical exponent