Multiple blowing-up solutions for asymptotically critical Lane-Emden systems on Riemannian manifolds

Let $(\mathcal{M},g)$ be a smooth compact Riemannian manifold of dimension $N\ge 8$. We are concerned with the following elliptic system \begin{align*} \left{ \begin{array}{ll} -Δ_g u+h(x)u=v^{p-α\varepsilon}, \ \ &\mbox{in}\ $\mathcal{M}$, -Δ_g v+h(x)v=u^{q-β\varepsilon}, \ \ &\mbox{in}\ $\mathcal{M}$, u,v>0, \ \ &\mbox{in}\ $\mathcal{M}$, \end{array} \right. \end{align*} where ${\mathrm{\Delta }}_g=di{v}_g\mathrm{\nabla }$ is the Laplace-Beltrami operator on $\mathcal{M}$, $h(x)$ is a $C^1$-function on $\mathcal{M}$, $\epsilon >0$ is a small parameter, $\alpha ,\beta >0$ are real numbers, $(p,q)\in (1,+\mathrm{\infty })\times (1,+\mathrm{\infty })$ 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.