In this paper, we study the following Lane–Emden system with non-power nonlinearity \(\begin{cases}-\Delta u={{\vert v\vert^{p-1}v}\over[\ln(e+\vert v\vert)]^{\epsilon}} & \text{in}\;\Omega,\\-\Delta v={\vert u\vert^{q-1}u\over[\ln(e+\vert u\vert)]^{\epsilon}} & \text{in}\;\Omega,\\u=v=0 & \text{on}\;\partial\Omega,\end{cases}\) where Ω is a bounded smooth domain in ℝN, N ≥ 3, ϵ > 0 is a small parameter, p and q lying on the critical Sobolev hyperbola \({1 \over {p + 1}} + {1 \over {q + 1}} = {{N - 2} \over N}\) . We construct multiple blowing-up solutions based on the finite dimensional Lyapunov–Schmidt reduction method as ϵ goes to zero.
