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 $$\begin{align}\label{yuanshi} \left\{ \begin{array}{ll} -\Delta_{g+\omega^2\kappa} u+h(x)u=v^{p-\alpha \varepsilon}, \ \ &\mbox{in $(\mathcal{M}\times_\omega \mathcal{K},g+\omega^2\kappa)$},\\ -\Delta_{g+\omega^2\kappa} v+h(x)v=u^{q-\beta \varepsilon}, \ \ &\mbox{in $(\mathcal{M}\times_\omega \mathcal{K},g+\omega^2\kappa)$},\\ u,v>0, \ \ &\mbox{in $(\mathcal{M}\times_\omega \mathcal{K},g+\omega^2\kappa)$}, \end{array} \right.\qquad(0.1)\end{align}$$ where $\Delta_{g+\omega^2\kappa} = div_{g+\omega^2\kappa} \nabla$ is the Laplace-Beltrami operator on $\mathcal{M}\times_\omega \mathcal{K}$, $h(x)$ is a $C^1$-function on $\mathcal{M}\times_\omega \mathcal{K}$, $\varepsilon>0$ is a small parameter, $\alpha,\beta>0$ are real numbers, $\varepsilon$ 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 $k\geq2$, using the Lyapunov-Schmidt reduction, we prove that problem (0.1) has a $k$-peaks solution concentrated along a $m$-dimensional minimal submanifold of $(\mathcal{M}\times_\omega \mathcal{K})^k$.