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Given a bounded domain \Omega\subseteq \RN, N\geq 2, and a positive integer m\in \N, consider the following optimal partition problem \inf{ \sum_{i=1}^m \lambdak(\omegai): \omegai \subset \Omega open \forall i, \omegai\cap \omegaj=\emptyset whenever i\neq j}, where \lambdak(\omega) denotes the k-th eigenvalue of -\Delta in H10(\omega). Approximating this problem by a system of elliptic equations with competition terms, we show the existence of regular optimal partitions. Moreover, multiplicity of sign-changing solutions for the approximating system is obtained.
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