Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911770177699840 |
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| author | Hou, Songbo Qiao, Wenjie |
| author_facet | Hou, Songbo Qiao, Wenjie |
| contents | Consider a finite connected graph denoted as $G=(V, E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$ Δu = λe^u (e^u - 1)^{2p+1} + f,$$ where $Δ$ denotes the graph Laplacian, $λ$ is a real number, $p$ is a non-negative integer, and $f$ is a function on $V$. Through the computation of the topological degree, this paper demonstrates the existence of a single solution for the model. Further analysis of the interplay between the topological degree and the critical group of an associated functional reveals the presence of multiple solutions. These findings extend the work of Li, Sun, Yang (arXiv:2309.12024) and Chao, Hou (J. Math. Anal. Appl. (2023) 126787). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_01990 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree Hou, Songbo Qiao, Wenjie Analysis of PDEs Functional Analysis 39A12, 46E39 Consider a finite connected graph denoted as $G=(V, E)$. This study explores a generalized Chern-Simons Higgs model, characterized by the equation: $$ Δu = λe^u (e^u - 1)^{2p+1} + f,$$ where $Δ$ denotes the graph Laplacian, $λ$ is a real number, $p$ is a non-negative integer, and $f$ is a function on $V$. Through the computation of the topological degree, this paper demonstrates the existence of a single solution for the model. Further analysis of the interplay between the topological degree and the critical group of an associated functional reveals the presence of multiple solutions. These findings extend the work of Li, Sun, Yang (arXiv:2309.12024) and Chao, Hou (J. Math. Anal. Appl. (2023) 126787). |
| title | Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree |
| topic | Analysis of PDEs Functional Analysis 39A12, 46E39 |
| url | https://arxiv.org/abs/2402.01990 |