Multiple mixed interior and boundary peaks synchronized solutions for linearly coupled Schrödinger systems

Abstract

In the present paper we consider the problem: \begin{equation} \label{0}\tag{N$_\varepsilon$} \begin{cases} -\varepsilon^{2}\Delta u+u=u^{3}+\lambda v& \text{in } \Omega, \\ -\varepsilon^{2}\Delta v+v=v^{3}+\lambda u& \text{in } \Omega,\\ u> 0,\ v> 0& \text{in } \Omega,\\ \dfrac{\partial u}{\partial n}=\dfrac{\partial v}{\partial n}=0& \text{on } \partial\Omega, \end{cases} \end{equation} where $\varepsilon> 0$, $0< \lambda< 1$, $\Omega\subset\mathbb{R}^{3}$ is smooth and bounded, and $n$ denotes the outer normal vector defined on $\partial\Omega$, the boundary of $\Omega$. By the Lyapunov-Schmidt reduction method and the maximum principle of elliptic equations, we construct synchronized solutions of (\ref{0}) with mixed interior and boundary peaks for any $0< \varepsilon< \varepsilon_0$ and $\lambda\in(0,1)\backslash\{\lambda_0\}$, where $\lambda_0\in(0,1)$ is given and $\varepsilon_0> 0$ is sufficiently small. As $\varepsilon$ approaches $0$, the interior peaks concentrate at sphere packing points in $\Omega$ and the boundary peaks concentrate at the critical points of the mean curvature function of the boundary.