Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EXISTENCE OF SOLUTIONS TO A GENERALIZED SELF-DUAL CHERN-SIMONS EQUATION ON FINITE GRAPHS

  • Yuanyang Hu (School of Mathematics and Statistics Henan University)
  • Received : 2023.05.16
  • Accepted : 2023.07.19
  • Published : 2024.01.01
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Abstract

Let G = (V, E) be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on G $${\Delta}u={\lambda}e^u(e^u-1)^5+4{\pi}\sum_{s=1}^{N}\delta_{ps}$$, where λ > 0, δps is the Dirac mass at the vertex ps, and p1, p2, . . . , pN are arbitrarily chosen distinct vertices on the graph. We show that there exists a critical value $\hat{\lambda}$ such that when λ > $\hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when λ = $\hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when λ < $\hat{\lambda}$, the generalized Chern-Simons equation has no solution.

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Acknowledgement

The author thanks the unknown referee very much for helpful suggestions. This work is financially supported by the Natural Science Foundation of Henan Province (Grant No. 222300 420416), the China Postdoctoral Science Foundation (Grant No. 2022M711045), and the National Natural Science Foundation of China (Grant No. 12201184).

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