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

Abstract

Let \((\mathcal {M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 8\). We are concerned with the following elliptic system

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _g u+h(x)u=v^{p-\alpha \varepsilon }, \ \ &{}\text{ in }\ \mathcal {M},\\ -\Delta _g v+h(x)v=u^{q-\beta \varepsilon }, \ \ &{}\text{ in }\ \mathcal {M},\\ u,v>0, \ \ &{}\text{ in }\ \mathcal {M}, \end{array} \right. \end{aligned}$$

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.