Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree
S Hou, W Qiao�- arXiv preprint arXiv:2402.01990, 2024 - arxiv.org
S Hou, W Qiao
arXiv preprint arXiv:2402.01990, 2024•arxiv.orgConsider a finite connected graph denoted as $ G=(V, E) $. This study explores a
generalized Chern-Simons Higgs model, characterized by the equation: $$\Delta u=\lambda
e^ u (e^ u-1)^{2p+ 1}+ f, $$ where $\Delta $ denotes the graph Laplacian, $\lambda $ 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�…
generalized Chern-Simons Higgs model, characterized by the equation: $$\Delta u=\lambda
e^ u (e^ u-1)^{2p+ 1}+ f, $$ where $\Delta $ denotes the graph Laplacian, $\lambda $ 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�…
Consider a finite connected graph denoted as . This study explores a generalized Chern-Simons Higgs model, characterized by the equation:
where denotes the graph Laplacian, is a real number, is a non-negative integer, and is a function on . 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).
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