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

Duong, Anh Tuan; Fujiié, Setsuro

Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1413-1423

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

$$v_t-\Delta v=v^{-p-2}-v^p,(x,t)\in V\times ℝ,$$

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

$$\left\{\begin{array}{c}{u}_t-\Delta u=u-u^3-\lambda u{v}^2,(x,t)\in V\times ℝ,\hfill \\ v_t-\Delta v=v-v^3-\lambda v{u}^2,(x,t)\in V\times ℝ,\hfill \end{array}\right.$$

where $p>0$, $\lambda >0$ and $\Delta$ is the standard discrete graph Laplacian. Firstly, we prove that any positive solution $v$ of the Lichnerowicz equation satisfies $v\ge 1$. 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 $\lambda$.

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

$$v_t-\Delta v=v^{-p-2}-v^p,(x,t)\in V\times ℝ,$$

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

$$\left\{\begin{array}{c}{u}_t-\Delta u=u-u^3-\lambda u{v}^2,(x,t)\in V\times ℝ,\hfill \\ v_t-\Delta v=v-v^3-\lambda v{u}^2,(x,t)\in V\times ℝ,\hfill \end{array}\right.$$

où $p>0$, $\lambda >0$ et $\Delta$ est le laplacien discret standard des graphes. Tout d’abord, nous prouvons que toutes les solutions positives de l’équation de Lichnerowicz satisfont à $v\ge 1$. 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 $\lambda$.

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

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