Multi-bump ground states of the Gierer–Meinhardt system in ℝ2

Manuel del Pino; Michał Kowalczyk; Juncheng Wei

doi:10.1016/s0294-1449(02)00024-0

JournalsaihpcVol. 20, No. 1pp. 53–85

Multi-bump ground states of the Gierer–Meinhardt system in ℝ2

Manuel del Pino
Departamento de Ingenierı́a Matemática y Centro de Modelamiento, Matemático Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
Michał Kowalczyk
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
Juncheng Wei
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

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Abstract

We consider the stationary Gierer–Meinhardt system in $R^{2}$ :

⎩ ⎨ ⎧ Δ A - A + \frac{A ^{2}}{H} = 0 Δ H - σ^{2} H + A^{2} = 0 A, H > 0; A, H \to 0 in R^{2}, in R^{2}, as ∣ x ∣ \to + \infty.

We construct multi-bump ground-state solutions for this system for all sufficiently small $σ$ . The centers of these bumps are located at the vertices of a regular polygon, while the bumps resemble, after a suitable scaling in their $A$ -coordinate, the unique radially symmetric solution of

{Δ w - w + w^{2} = 0 0 < w (y) \to 0 in R^{2}, as ∣ y ∣ \to \infty.

A similar construction is made for vertices of two concentric polygons and a general procedure for detection of organized finite patterns is suggested.

Résumé

On considère le système stationnaire de Gierer–Meinhardt dans

⎩ ⎨ ⎧ Δ A - A + \frac{A ^{2}}{H} = 0 Δ H - σ^{2} H + A^{2} = 0 A, H > 0; A, H \to 0 dans R^{2}, dans R^{2}, quand ∣ x ∣ \to + \infty.

On construit des solutions fondamentales multi-pics pour ce système, pour tout $σ$ suffisamment petit. Les centres de ces pics sont localisés aux sommets d’un polygone régulier, tandis que les pics ressemblent, après un changement d’échelle approprié dans la coordonnée $A$ , à l’unique solution radiale de

{Δ w - w + w^{2} = 0 0 < w (y) \to 0 dans R^{2}, quand ∣ y ∣ \to \infty.

Une construction similaire est faite pour les sommets de deux polygones concentriques et une procédure générale de détection destructures finies organisées est suggérée.

Cite this article

Manuel del Pino, Michał Kowalczyk, Juncheng Wei, Multi-bump ground states of the Gierer–Meinhardt system in ℝ2. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 1, pp. 53–85

DOI 10.1016/S0294-1449(02)00024-0