Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems

We study the singularly perturbed system of elliptic equations
\begin{equation}\label{P_e}
\begin{cases}
-\epsilon^2\Delta u_i+u_i=\mu_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ell\lambda_{ij}\beta_{ij}|u_j|^{\alpha_{ij}}|u_i|^{\beta_{ij} -2}u_i,\\
u_i \in H^1_0(\Omega), \quad u_i\neq 0, \qquad i=1,\ldots,\ell,
\end{cases}
\end{equation}
in a bounded domain \(\Omega\) in \(\mathbb{R}^N\), with \(N\geq 3\), \(\epsilon>0\), \(\mu_i>0\), \(\lambda_{ij}=\lambda_{ji}<0\), \(\alpha_{ij}, \beta_{ij}>1\), \(\alpha_{ij}=\beta_{ji}\), \(\alpha_{ij} + \beta_{ij} = p\in (2,2^*)\), and \(2^{*}:=\frac{2N}{N-2}\). If \(\Omega\) is the unit ball, we obtain solutions with a prescribed combination of positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as \(\epsilon\to 0\): solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components of the limit system
\begin{equation}\label{Sl_i}
\begin{cases}
-\Delta u_i+u_i=\mu_i|u_i|^{p-2}u_i + \sum\limits_{\substack{j=1 \\ j \not=i}}^\ell\lambda_{ij}\beta_{ij}|u_j|^{\alpha_{ij}}|u_i|^{\beta_{ij} -2}u_i,\\
u_i \in H^1(\mathbb{R}^N), \quad u_i\neq 0, \qquad i=1,\ldots,\ell,
\end{cases}
\end{equation}
and solutions whose limit profile is a solution of the uncoupled system, i.e., after rescaling and translation, the limit profile of the \(i\)-th component is a positive or a nonradial sign-changing solution to the to the problem
\begin{eqnarray}\label{P_i}
-\Delta u+u=\mu_i|u|^{p-2}u,\quad \quad
u \in H^1(\mathbb{R}^N), \quad u\neq 0.
\end{eqnarray}