Sign-changing solutions of competition–diffusion elliptic systems and optimal partition problems
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 279-300.

Dans cet article nous montrons lʼexistence dʼune infinité de solutions qui changent de signe pour le système dʼéquations de Schrödinger avec des interactions compétitives

-Δu i +a i u i 3 +βu i ji u j 2 =λ i,β u i ,u i H 0 1 (Ω),i=1,,m
Ω est un domaine borné, β>0 et a i 0i. De plus, quand a i =0, nous démontrons une relation entre les énergies critiques associées à ce système et le problème de partition optimale
inf ω i Ωopen ω i ω j =ij i=1 m λ k i (ω i ),
λ k i (ω) indiques la k i -ème valeur propre de lʼopérateur −Δ in H 0 1 (ω). Dans le cas k i 2, nous montrons que le problème de partition optimale apparaît comme une valeur limite critique, en tant que paramètre de compétition β diverge vers +∞.

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 ji u j 2 =λ i,β u i ,u i H 0 1 (Ω),i=1,,m
where Ω is a bounded domain, β>0 and a i 0i. Moreover, for a i =0, we show a relation between critical energies associated with this system and the optimal partition problem
inf ω i Ωopen ω i ω j =ij i=1 m λ k i (ω i ),
where λ k i (ω) denotes the k i -th eigenvalue of −Δ in H 0 1 (ω). In the case k i 2 we show that the optimal partition problem appears as a limiting critical value, as the competition parameter β diverges to +∞.

DOI : 10.1016/j.anihpc.2011.10.006
Mots clés : Elliptic systems, Optimal partition problems, Sign-changing solutions, Minimax methods
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     title = {Sign-changing solutions of competition{\textendash}diffusion elliptic systems and optimal partition problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {279--300},
     publisher = {Elsevier},
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Tavares, Hugo; Terracini, Susanna. Sign-changing solutions of competition–diffusion elliptic systems and optimal partition problems. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 279-300. doi : 10.1016/j.anihpc.2011.10.006. http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.006/

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