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

Tavares, Hugo; Terracini, Susanna

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 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

$$-\Delta u_i+a_i{u}_i^3+\beta u_i\sum _{j\ne i}{u}_j^2={\lambda }_{i,\beta }{u}_i,u_i\in H_0^1\left(\Omega \right),i=1,\cdots ,m$$

where Ω is a bounded domain, $\beta >0$ 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

$$\underset{\genfrac{}{}{0.0pt}{}{{\omega }_i\subset \Omega \text{open}}{{\omega }_i\cap {\omega }_j=⌀\forall i\ne j}}{\inf }\sum _{i=1}^m{\lambda }_{k_i}\left({\omega }_i\right),$$

where ${\lambda }_{k_i}\left(\omega \right)$ denotes the $k_i$-th eigenvalue of −Δ in $H_0^1\left(\omega \right)$. 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

$$-\Delta u_i+a_i{u}_i^3+\beta u_i\sum _{j\ne i}{u}_j^2={\lambda }_{i,\beta }{u}_i,u_i\in H_0^1\left(\Omega \right),i=1,\cdots ,m$$

où Ω est un domaine borné, $\beta >0$ 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

$$\underset{\genfrac{}{}{0.0pt}{}{{\omega }_i\subset \Omega \text{open}}{{\omega }_i\cap {\omega }_j=⌀\forall i\ne j}}{\inf }\sum _{i=1}^m{\lambda }_{k_i}\left({\omega }_i\right),$$

où ${\lambda }_{k_i}\left(\omega \right)$ indiques la $k_i$-ème valeur propre de lʼopérateur −Δ in $H_0^1\left(\omega \right)$. 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 +∞.

Keywords: Elliptic systems, Optimal partition problems, Sign-changing solutions, Minimax methods

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