Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs

2022/02/05 by Yuanyang Hu, Hu, Yuanyang · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2202.02525

openalex publication_date 2022/02/05 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

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 Δu=λe^u(e^u-1)⁵+4 π∑_s=1^N δ_p_s , where λ>0, δ_p_s is the Dirac mass at the vetex p_s, and p₁, p₂,…, p_N are arbitrarily chosen distinct vertices on the graph. We show that there exists a critial value λ such that when λ> λ, the generalized Chern-Simons equation has at least two solutions, when λ= λ, the generalized Chern-Simons equation has a solution, and when λ< λ, the generalized Chern-Simons equation has no solution.

Cited by

• Topological solutions of generalized Chern-Simons equations on discrete lattice graphs
• Existence theory for elliptic equations of general exponential nonlinearity on finite graphs

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