Abstract
In this paper, we consider a relativistic Abelian Chern–Simons equation
on a connected finite graph \(G=(V, E)\), where \(\lambda >0\) is a constant; \(a>b>0\); \(N_{1}\) and \(N_{2}\) are positive integers; \(p_{1}, p_{2}, \ldots , p_{N_{1}}\) and \(q_{1}, q_{2}, \ldots , q_{N_{2}}\) denote distinct vertices of V. Additionally, \(\delta _{p_{j}}\) and \(\delta _{q_{j}}\) represent the Dirac delta masses located at vertices \(p_{j}\) and \(q_{j}\). By employing the method of constrained minimization, we prove that there exists a critical value \(\lambda _{0}\), such that the above equation admits a solution when \(\lambda \ge \lambda _{0}\). Furthermore, we employ the mountain pass theorem developed by Ambrosetti–Rabinowitz to establish that the equation has at least two solutions when \(\lambda >\lambda _{0}\).
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Communicated by Behzad Djafari-Rouhani.
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Zhao, J. A Relativistic Abelian Chern–Simons Model on Graph. Bull. Iran. Math. Soc. 49, 89 (2023). https://doi.org/10.1007/s41980-023-00830-3
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DOI: https://doi.org/10.1007/s41980-023-00830-3