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In this paper we prove the existence of infinitely many sign-changingsolutions for the system of $m$ Schr\"odinger equations with competitioninteractions  $$ -\Delta u_i+a_i u_i^3+\beta u_i \sum_{j\neq i} u_j^2 =\lambda_{i,\beta}u_i \quad u_i\in H^1_0(\Omega), \quad i=1,...,m $$  where $\Omega$ is a bounded domain, $\beta>0$ and $a_i\geq 0\ \forall i.$Moreover, for $a_i=0$, we show a relation between critical energies associatedwith this system and the optimal partition problem  $$ \mathop{\inf_{\omega_i\subset \Omega \text{open}}}_{\omega_i\cap\omega_j=\emptyset\forall i\neq j} \sum_{i=1}^{m} \lambda_{k_i}(\omega_i), $$  where $\lambda_{k_i}(\omega)$ denotes the $k_i$--th eigenvalue of $-\Delta$in $H^1_0(\omega)$. In the case $k_i\leq 2$ we show that the optimal partitionproblem appears as a limiting critical value, as the competition parameter$\beta$ diverges to $+\infty$.

Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems