Sign-changing solutions of competition–diffusion elliptic systems and optimal partition problems

Tavares, Hugo; Terracini, Susanna

Tavares, Hugo; Terracini, Susanna

Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, p. 279-300

Abstract
Résumé

In this paper we prove the existence of infinitely many sign-changing solutions for the system of m Schrödinger equations with competition interactions $- Δ u_{i} + a_{i} u_{i}^{3} + β u_{i} \sum_{j \neq i} u_{j}^{2} = λ_{i, β} u_{i}, u_{i} \in H_{0}^{1} (Ω), i = 1, \dots, m$ where Ω is a bounded domain, $β > 0$ and $a_{i} ⩾ 0$ ∀i. Moreover, for $a_{i} = 0$ , we show a relation between critical energies associated with this system and the optimal partition problem $\inf_{\binom{ω_{i} \subset Ω open}{ω_{i} \cap ω_{j} = ⌀ \forall i \neq j}} \sum_{i = 1}^{m} λ_{k_{i}} (ω_{i}),$ where $λ_{k_{i}} (ω)$ denotes the $k_{i}$ -th eigenvalue of −Δ in $H_{0}^{1} (ω)$ . 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 +∞.

Dans cet article nous montrons lʼexistence dʼune infinité de solutions qui changent de signe pour le système dʼéquations de Schrödinger avec des interactions compétitives $- Δ u_{i} + a_{i} u_{i}^{3} + β u_{i} \sum_{j \neq i} u_{j}^{2} = λ_{i, β} u_{i}, u_{i} \in H_{0}^{1} (Ω), i = 1, \dots, m$ où Ω est un domaine borné, $β > 0$ et $a_{i} ⩾ 0$ ∀i. De plus, quand $a_{i} = 0$ , nous démontrons une relation entre les énergies critiques associées à ce système et le problème de partition optimale $\inf_{\binom{ω_{i} \subset Ω open}{ω_{i} \cap ω_{j} = ⌀ \forall i \neq j}} \sum_{i = 1}^{m} λ_{k_{i}} (ω_{i}),$ où $λ_{k_{i}} (ω)$ 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 +∞.

MR 2901198 | Zbl 1241.35046 | 1 citation in Numdam

DOI : https://doi.org/10.1016/j.anihpc.2011.10.006
Keywords: Elliptic systems, Optimal partition problems, Sign-changing solutions, Minimax methods

@article{AIHPC_2012__29_2_279_0,
     author = {Tavares, Hugo and Terracini, Susanna},
     title = {Sign-changing solutions of competition--diffusion elliptic systems and optimal partition problems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {29},
     number = {2},
     year = {2012},
     pages = {279-300},
     doi = {10.1016/j.anihpc.2011.10.006},
     zbl = {1241.35046},
     mrnumber = {2901198},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2012__29_2_279_0}
}

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, Volume 29 (2012) no. 2, pp. 279-300. doi : 10.1016/j.anihpc.2011.10.006. http://www.numdam.org/item/AIHPC_2012__29_2_279_0/

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