Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs
Let $(V,E)$ be a locally finite weighted graph. We study some qualitative properties of positive solutions of the Lichnerowicz equation \[ v_t-\Delta v=v^{-p-2}-v^p, \;(x,t)\in V \times \mathbb{R}, \] and of (sign-changing) solutions of the Ginzburg-Landau system \[ {\left\lbrace \begin{array}{ll} u...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-11-01
|
| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.653/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206146776956928 |
|---|---|
| author | Duong, Anh Tuan Fujiié, Setsuro |
| author_facet | Duong, Anh Tuan Fujiié, Setsuro |
| author_sort | Duong, Anh Tuan |
| collection | DOAJ |
| description | Let $(V,E)$ be a locally finite weighted graph. We study some qualitative properties of positive solutions of the Lichnerowicz equation
\[ v_t-\Delta v=v^{-p-2}-v^p, \;(x,t)\in V \times \mathbb{R}, \]
and of (sign-changing) solutions of the Ginzburg-Landau system
\[ {\left\lbrace \begin{array}{ll} u_t-\Delta u=u-u^3-\lambda uv^2 , \;(x,t)\in V \times \mathbb{R},\\ v_t-\Delta v=v-v^3-\lambda vu^2, \;(x,t)\in V \times \mathbb{R}, \end{array}\right.} \]
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 $. |
| format | Article |
| id | doaj-art-4952b899c92c4ff3912c382d2803e267 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-4952b899c92c4ff3912c382d2803e2672025-02-07T11:23:50ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111413142310.5802/crmath.65310.5802/crmath.653Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphsDuong, Anh Tuan0Fujiié, Setsuro1Faculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam.Department of Mathematical Sciences, Ritsumeikan University, JapanLet $(V,E)$ be a locally finite weighted graph. We study some qualitative properties of positive solutions of the Lichnerowicz equation \[ v_t-\Delta v=v^{-p-2}-v^p, \;(x,t)\in V \times \mathbb{R}, \] and of (sign-changing) solutions of the Ginzburg-Landau system \[ {\left\lbrace \begin{array}{ll} u_t-\Delta u=u-u^3-\lambda uv^2 , \;(x,t)\in V \times \mathbb{R},\\ v_t-\Delta v=v-v^3-\lambda vu^2, \;(x,t)\in V \times \mathbb{R}, \end{array}\right.} \] 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 $.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.653/Liouville-type theoremsLichnerowicz equationsGinzburg–Landau systemnonexistence resultsqualitative propertylocally finite graphs |
| spellingShingle | Duong, Anh Tuan Fujiié, Setsuro Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs Comptes Rendus. Mathématique Liouville-type theorems Lichnerowicz equations Ginzburg–Landau system nonexistence results qualitative property locally finite graphs |
| title | Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs |
| title_full | Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs |
| title_fullStr | Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs |
| title_full_unstemmed | Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs |
| title_short | Some qualitative properties of Lichnerowicz equations and Ginzburg–Landau systems on locally finite graphs |
| title_sort | some qualitative properties of lichnerowicz equations and ginzburg landau systems on locally finite graphs |
| topic | Liouville-type theorems Lichnerowicz equations Ginzburg–Landau system nonexistence results qualitative property locally finite graphs |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.653/ |
| work_keys_str_mv | AT duonganhtuan somequalitativepropertiesoflichnerowiczequationsandginzburglandausystemsonlocallyfinitegraphs AT fujiiesetsuro somequalitativepropertiesoflichnerowiczequationsandginzburglandausystemsonlocallyfinitegraphs |