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

In this paper we prove the existence of infinitely many sign-changing solutions for the system of $m$ Schr\"odinger equations with competition interactions $$-\Delta u_i+a_i u_i^3+\beta u_i \sum_{j\neq i} u_j^2 =\lambda_{i,\beta} u_i \quad u_i\in H^1_0(\Omega), \quad i=1,...,m$$ 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 associated with 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 partition problem appears as a limiting critical value, as the competition parameter $\beta$ diverges to $+\infty$.