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| Existence of Solutions to a Generalized Self-dual Chern-Simons Equation on Graphs |
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Citation： |
Yingshu LÜ,Peirong ZHONG.Existence of Solutions to a Generalized Self-dual Chern-Simons Equation on Graphs[J].Chinese Annals of Mathematics B,2026,(3):429~448 |
| Page view： 112
Net amount： 49 |
Authors： |
Yingshu Lü; Peirong ZHONG |
Foundation： |
National Natural Science Foundation of China (Nos.12401263, 12571225, 12141105) and the Shanghai Frontiers Science Center of Modern Analysis. |
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| Abstract： |
Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. In this paper, the authors consider a generalized self-dual Chern-Simons equation on the graph G: Δu=?λeF(u)[eF(u)?1]2+4πΣj=1Mδpj, where eF(u) satisfies u=1+F(u)?eF(u), F(u)=F(u), u≤0; (0, u>0), λ>0, M is any fixed positive integer, δpj is the Dirac delta mass at the vertex pj, and p1, p2, ···, pM are arbitrarily chosen distinct vertices on the graph. They first prove that there is a critical value λc such that if λ ≥ λc, then the generalized self-dual Chern-Simons equation has a solution uλ. Applying the existence result, they develop a new method to construct a solution of (0.1) which is monotonic with respect to λ when λ ≥ λc. Then they establish that there exist at least two solutions of the equation for λ>λc via the variational method. Furthermore, they give a fine estimate of the monotone solution, which can be applied to other related problems. |
Keywords： |
Chern-Simons equation Finite graph Existence Variational method |
Classification： |
35A01, 35A15, 35R02 |
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