  Let $(\mathcal{M},g)$ be a smooth compact Riemannian manifold of dimension
$N\geq 8$. We are concerned with the following elliptic system \begin{align*}
  \left\{
  \begin{array}{ll}
  -\Delta_g u+h(x)u=v^{p-\alpha \varepsilon},
  \ \ &\mbox{in}\ \mathcal{M},
  -\Delta_g v+h(x)v=u^{q-\beta \varepsilon},
  \ \ &\mbox{in}\ \mathcal{M},
  u,v>0,
  \ \ &\mbox{in}\ \mathcal{M},
  \end{array}
  \right.
  \end{align*} where $\Delta _g=div_g \nabla$ is the Laplace-Beltrami operator
on $\mathcal{M}$, $h(x)$ is a $C^1$-function on $\mathcal{M}$, $\varepsilon>0$
is a small parameter, $\alpha,\beta>0$ are real numbers, $(p,q)\in
(1,+\infty)\times (1,+\infty)$ satisfies
$\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$. Using the Lyapunov-Schmidt
reduction method, we obtain the existence of multiple blowing-up solutions for
the above problem.
