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.
Publisher
Uniwersytet Mikolaja Kopernika/Nicolaus Copernicus University