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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 $m$ Schrödinger equations with competition interactions $$ -Δu_i+a_i u_i^3+βu_i \sum_{j\neq i} u_j^2 =λ_{i,β} u_i \quad u_i\in H^1_0(Ω), \quad i=1,...,m $$ where $Ω$ is a bounded domain, $β>0$ 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 $$ \mathop{\inf_{ω_i\subset Ω\text{open}}}_{ω_i\cap ω_j=\emptyset\forall i\neq j} \sum_{i=1}^{m} λ_{k_i}(ω_i), $$ where $λ_{k_i}(ω)$ denotes the $k_i$--th eigenvalue of $-Δ$ in $H^1_0(ω)$. In the case $k_i\leq 2$ we show that the optimal partition problem appears as a limiting critical value, as the competition parameter $β$ diverges to $+\infty$.