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 Schrodinger equations with competition interactions -Delta u(i) + a(i)u(i)(3) + beta u(i) (j not equal i)Sigma u(2)j = lambda(i),beta u(i), u(i) is an element of H-0(1)(Omega), i = 1, ... , m where Omega is a bounded domain, beta > 0 and (a(i) >= 0 for all i. Moreover, for a(i) = O. we show a relation between critical energies associated with this system and the optimal partition problem omega i boolean AND omega j=(sic)for all i not equal j omega i subset of Omega open inf (i=1)Sigma(m) lambda(ki)(omega(i)), where lambda(ki) (omega) denotes the k(i)-th eigenvalue of -Delta in H-0(1) (omega). In the case k(i) <= 2 we show that the optimal partition problem appears as a limiting critical value, as the competition parameter beta diverges to + infinity. (C) 2011 Elsevier Masson SAS. All rights reserved.