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, p. 279-300
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 ∑ j≠i 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 ⩾0 ∀i. 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 =⌀∀i≠j∑ 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 -Δu i +a i u i 3 +βu i ∑ j≠i u j 2 =λ i,β u i ,u i ∈H 0 1 (Ω),i=1,⋯,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 ω i ⊂Ωopen ω i ∩ω j =⌀∀i≠j∑ 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 +∞.
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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