  In this paper, we study the following Lane-Emden system with nearly critical
non-power nonlinearity \begin{eqnarray*} \left\{ \arraycolsep=1.5pt
  \begin{array}{lll} -\Delta u =\frac{|v|^{p-1}v}{[\ln(e+|v|)]^\epsilon}\ \
&{\rm in}\ \Omega, \\[2mm] -\Delta v =\frac{|u|^{q-1}u}{[\ln(e+|u|)]^\epsilon}\
\ &{\rm in}\ \Omega, \\[2mm] u= v=0 \ \ & {\rm on}\ \partial\Omega, \end{array}
\right. \end{eqnarray*} where $\Omega$ is a bounded smooth domain in
$\mathbb{R}^N$, $N\geq 3$, $\epsilon>0$ is a small parameter, $p$ and $q $
lying on the critical Sobolev hyperbola
$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$. We construct multiple blowing-up
solutions based on the finite dimensional Lyapunov-Schmidt reduction method as
$\epsilon$ goes to zero.
