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We study the existence and asymptotic behavior of solutions having positiveand sign-changing components to the singularly perturbed system of ellipticequations \begin{equation*} \begin{cases} -\varepsilon^2\Deltau_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 $N\geq 4$, $\varepsilon>0$, $\mu_i>0$, $\lambda_{ij}=\lambda_{ji}<0$,$\alpha_{ij}, \beta_{ij}>1$, $\alpha_{ij}=\beta_{ji}$, $\alpha_{ij} +\beta_{ij} = p\in (2,2^*)$, and $2^{*}:=\frac{2N}{N-2}$.  If $\Omega$ is the unit ball we obtain solutions with a prescribedcombination of positive and nonradial sign-changing components exhibiting twodifferent types of asymptotic behavior as $\varepsilon\to 0$: solutions whoselimit profile is a rescaling of a solution with positive and nonradialsign-changing components of the limit system \begin{equation*} \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 ofthe uncoupled system, i.e., after rescaling and translation, the limit profileof the $i$-th component is a positive or a nonradial sign-changing solution tothe equation $$-\Delta u+u=\mu_i|u|^{p-2}u,\qquad u \in H^1(\mathbb{R}^N),\qquad u\neq 0.$$

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