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 $$-\mathrm{\Delta }{u}_i+a_i{u}_i^3+\beta u_i\sum _{j\mathrm{\ne }i}{u}_j^2={\lambda }_{i,\beta }{u}_i{u}_i\in H_0^1(\mathrm{\Omega }),i=1,\mathrm{.}\mathrm{.}\mathrm{.},m$$ where $\mathrm{\Omega }$ is a bounded domain, $\beta >0$ and $a_i\ge 0\mathrm{\forall }i\mathrm{.}$ Moreover, for $a_i=0$, we show a relation between critical energies associated with this system and the optimal partition problem $${}_{{\omega }_i\cap {\omega }_j=\mathrm{\varnothing }\mathrm{\forall }i\mathrm{\ne }j}\sum _{i=1}^m{\lambda }_{k_i}({\omega }_i),$$ where ${\lambda }_{k_i}(\omega )$ denotes the $k_i$–th eigenvalue of $-\mathrm{\Delta }$ in $H_0^1(\omega )$. In the case $k_i\le 2$ we show that the optimal partition problem appears as a limiting critical value, as the competition parameter $\beta$ diverges to $+\mathrm{\infty }$.