Authors’ abstract: We study the following elliptic system with critical exponent: \[\begin{cases} - \Delta u = \lambda_1u + u_1|u|^{2*-2}u + \beta |u|^{\frac{2*}{2} - 2}u|v|^{\frac{2*}{2}},\quad & x \in \Omega \\ - \Delta v = \lambda_2v + u_2|v|^{2*-2}v + \beta |v|^{\frac{2*}{2} - 2}v|u|^{\frac{2*}{2}},\quad & x \in \Omega \\ u = v = 0,\quad & x \in \partial \Omega, \end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(N=5\), \(2*:=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\mu_1,\mu_2 > 0\), \(\beta \in (- \sqrt{\mu_1,\mu_2},0), 0 <\lambda_{1}, \lambda_{2} < \lambda_{1} (\Omega), \lambda_1 (\Omega)\) is the first eigenvalue of \(-\Delta\) in \(H^1_0(\Omega)\). In [Commun. Partial Differ. Equations 39, No. 10, 1827–1859 (2014; Zbl 1308.35084)], Z. Chen et al. established a sign-changing solution of the above system in the case \(N\geq 6\) for \(\beta < 0\) and \(\lambda_{1},\lambda_{2} \in (0, \lambda_1(\Omega))\). We show that in dimension \(N = 5\), for \(\lambda_{1}\) and \(\lambda_{2}\) slightly smaller than \(\lambda_{1}(\Omega)\), the above system has a sign-changing solution in the following sense: one component changes sign and has exactly two nodal domains, while the other one is positive.