Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs

Anh Tuan Duong; Setsuro Fujiié

doi:10.5802/crmath.653

Article de recherche - Équations aux dérivées partielles

Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs
[Quelques propriétés qualitatives des équations de Lichnerowicz et de Ginzburg–Landau sur les graphes localement finis]

Anh Tuan Duong ¹ ; Setsuro Fujiié ²

¹ Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam.
² Department of Mathematical Sciences, Ritsumeikan University, Japan

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1413-1423.

Résumé
Abstract

Soit $(V, E)$ un graphe pondéré localement fini. Nous étudions certaines propriétés qualitatives des solutions positives de l’équation de Lichnerowicz

v_{t} - Δ v = v^{- p - 2} - v^{p}, (x, t) \in V \times ℝ,

et des solutions (avec changement de signe) du système de Ginzburg-Landau

\{\begin{matrix} u_{t} - Δ u = u - u^{3} - λ u v^{2}, (x, t) \in V \times ℝ, \\ v_{t} - Δ v = v - v^{3} - λ v u^{2}, (x, t) \in V \times ℝ, \end{matrix}

où $p > 0$ , $λ > 0$ et $Δ$ est le laplacien discret standard des graphes. Tout d’abord, nous prouvons que toutes les solutions positives de l’équation de Lichnerowicz satisfont à $v \geq 1$ . De plus, si nous supposons qu’une solution positive v est bornée, alors elle doit être triviale, c’est-à-dire $v \equiv 1$ . Nous construisons également une solution positive non bornée de l’équation de Lichnerowicz. Deuxièmement, nous obtenons une majoration précise pour les solutions du système de Ginzburg-Landau en fonction de l’étendue $λ$ .

Let $(V, E)$ be a locally finite weighted graph. We study some qualitative properties of positive solutions of the Lichnerowicz equation

v_{t} - Δ v = v^{- p - 2} - v^{p}, (x, t) \in V \times ℝ,

and of (sign-changing) solutions of the Ginzburg-Landau system

\{\begin{matrix} u_{t} - Δ u = u - u^{3} - λ u v^{2}, (x, t) \in V \times ℝ, \\ v_{t} - Δ v = v - v^{3} - λ v u^{2}, (x, t) \in V \times ℝ, \end{matrix}

where $p > 0$ , $λ > 0$ and $Δ$ is the standard discrete graph Laplacian. Firstly, we prove that any positive solution $v$ of the Lichnerowicz equation satisfies $v \geq 1$ . Moreover, if we assume the boundedness of positive solution $v$ , then it must be trivial, i.e $v \equiv 1$ . We also construct a nontrivial positive solution of the Lichnerowicz equation to show that the boundedness assumption is necessary. Secondly, we obtain sharp upper bound for solutions of the Ginzburg-Landau system depending on the range of $λ$ .

Reçu le : 2023-07-23
Révisé le : 2024-05-31
Accepté le : 2024-06-01
Publié le : 2024-11-14

DOI : 10.5802/crmath.653

Classification : 35B02, 35B45, 05C09
Keywords: Liouville-type theorems, Lichnerowicz equations, Ginzburg–Landau system, nonexistence results, qualitative property, locally finite graphs
Mot clés : Théorèmes de type Liouville, équations de Lichnerowicz, système de Ginzburg–Landau, résultats de non-existence, propriété qualitative, graphes localement finis

Affiliations des auteurs :

Anh Tuan Duong ¹ ; Setsuro Fujiié ²

¹ Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam.
² Department of Mathematical Sciences, Ritsumeikan University, Japan

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Anh Tuan Duong and Setsuro Fujii\'e},
     title = {Some qualitative properties of {Lichnerowicz} equations and {Ginzburg{\textendash}Landau} systems on locally finite graphs},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1413--1423},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.653},
     language = {en},
}

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Anh Tuan Duong; Setsuro Fujiié. Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1413-1423. doi : 10.5802/crmath.653. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.653/

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