Multiple Blowing-Up Solutions for Asymptotically Critical Lane-Emden Systems on Riemannian Manifolds

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