Let \(\Omega\) be a bounded domain in \(\mathbb R^n\) (\(n\geq 4\)) with smooth boundary, \(\lambda, \mu\in\mathbb R\), and \(p\geq q>1\). The authors investigate the solvability of the system \[ -\Delta v=\lambda u+| u|^{p-1}u,\quad -\Delta u=\mu v+| v|^{q-1}v\quad \text{in } \Omega,\qquad u=0=v\quad\text{on }\partial\Omega, \] in case that \({1\over{p+1}}+{1\over{q+1}}={{n-2}\over n}\) (critical Sobolev exponents). They show the existence of a positive solution (\(u>0\), \(v>0\)) provided that \(\lambda\) and \(\mu\) are positive and \(\lambda \mu<\lambda_1^2\) with \(\lambda_1\) the principal eigenvalue for the Laplacian on \(\Omega\) under homogeneous Dirichlet conditions. This extends a well-known result of Brezis and Nirenberg for the scalar problem \[ -\Delta u=\lambda u+| u|^{p-1}u\quad \text{in } \Omega, \qquad u=0 \quad\text{on }\partial\Omega \] (\(p=(n+2)/(n-2)\)) which asserts positive solvability if \(0<\lambda<\lambda_1\). The authors also provide conditions under which the system has nontrivial, not necessarily positive solutions. The proof employs a variational approach and relies on a dual formulation due to Clarke and Ekeland, which was already utilized by Ambrosetti and Struwe as an alternative to Brezis and Nirenberg’s original proof. The advantage is that the critical points of the dual functionals are mountain passes.