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引用次数: 0
摘要
考虑一个有限连通图，表示为 G = (V，E)。本研究探讨了一个广义的切尔恩-西蒙斯希格斯模型，该模型的方程Δu=λeu(eu-1)2p+1+f，其中Δ表示图的拉普拉奇，λ是实数，p是非负整数，f是 V 上的函数。通过计算拓扑度，本文证明了该模型存在单解。通过进一步分析拓扑度和相关函数临界群之间的相互作用，发现了多解的存在。这些发现扩展了 Li 等人 [Calc. Var. 63, 81 (2024)] 以及 Chao 和 Hou [J. Math. Anal. Appl. 519, 126787 (2023)] 的工作。
Solutions to a generalized Chern–Simons Higgs model on finite graphs by topological degree
Consider a finite connected graph denoted as G = (V, E). This study explores a generalized Chern-Simons Higgs model, characterized by the equation Δu=λeu(eu−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 et al. [Calc. Var. 63, 81 (2024)] and Chao and Hou [J. Math. Anal. Appl. 519, 126787 (2023)].
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