Let \(\Omega\) be a bounded smooth domain in \(\mathbb R^N\), \(N\geq 4\) and \(\lambda,\mu\in\mathbb R\). The author deals with the following problem: \[ -\Delta v=\lambda u+K(x)|u|^{p-1}u\text{ in }\Omega, \quad -\Delta u=\mu v+ Q(x)|v|^{q-1}v\text{ in }\Omega, \quad u=v=0\text{ on }\partial\Omega, \tag{1} \] where \(p,q>1\) and coefficients \(K,Q\) are positive and continuous functions on \(\overline{\Omega}\). The system (1) is studied in the case of critical exponents. To this end they use dual variational method. Moreover, under some natural assumptions on \(K(x)\) and \(Q(x)\) they prove the existence of nontrivial solutions for (1).