Blow-up solutions concentrated along minimal submanifolds for asymptotically critical Lane-Emden systems on Riemannian manifolds

Abstract

Let $(\mathcal{M},g)$ and $(\mathcal{K},\kappa )$ be two Riemannian manifolds of dimensions $N$ and $m$, respectively. Let $\omega \in C^2(\mathcal{M})$, $\omega >0$. The warped product $\mathcal{M}{\times }_{\omega }\mathcal{K}$ is the $(N+m)$-dimensional product manifold $\mathcal{M}\times \mathcal{K}$ furnished with metric $g+{\omega }^2\kappa$. We are concerned with the following elliptic system $$\begin{align}\label{yuanshi} \left\{ \begin{array}{ll} -Δ_{g+ω^2κ} u+h(x)u=v^{p-α\varepsilon}, \ \ &\mbox{in $(\mathcal{M}\times_ω\mathcal{K},g+ω^2κ)$},\\ -Δ_{g+ω^2κ} v+h(x)v=u^{q-β\varepsilon}, \ \ &\mbox{in $(\mathcal{M}\times_ω\mathcal{K},g+ω^2κ)$},\\ u,v>0, \ \ &\mbox{in $(\mathcal{M}\times_ω\mathcal{K},g+ω^2κ)$}, \end{array} \right.\qquad(0.1)\end{align}$$ where ${\Delta }_{g+{\omega }^2\kappa }=di{v}_{g+{\omega }^2\kappa }\nabla$ is the Laplace-Beltrami operator on $\mathcal{M}{\times }_{\omega }\mathcal{K}$, $h(x)$ is a $C^1$-function on $\mathcal{M}{\times }_{\omega }\mathcal{K}$, $\epsilon >0$ is a small parameter, $\alpha ,\beta >0$ are real numbers, $\epsilon$ is a positive parameter, $(p,q)\in (1,+\infty )\times (1,+\infty )$ satisfies $\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}$. For any given integer $k\ge 2$, using the Lyapunov-Schmidt reduction, we prove that problem (0.1) has a $k$-peaks solution concentrated along a $m$-dimensional minimal submanifold of $(\mathcal{M}{\times }_{\omega }\mathcal{K}{)}^k$.