2021/07/27 by Yingshu Lü, Lü, Yingshu, Peirong Zhong +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2107.12535
Let G=(V,E) be a connected finite graph and Δ the usual graph Laplacian. In this paper, we consider a generalized self-dual Chern-Simons equation on the graph G Δu=-λeF(u)[eF(u)-1]2+4π∑i=1Mδ_pj, where F(u)=\\widetildeF(u), u≤0, 0, u · gt;0, . \widetildeF(u) satisfies u=1+\widetilde F(u)-e^\widetilde F(u) , λ>0 , M is any fixed positive integer, δ_pj is the Dirac delta mass at the vertex pj, and p1, p2, ⋯, pM are arbitrarily chosen distinct vertices on the graph. We 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, we develop a new method to construct a solution of the equation which is monotonic with respect to λ when λ≥λc. Then we establish that there exist at least two solutions of the equation via the variational method for λ>λc. Furthermore, we give a fine estimate of the monotone solution which can be applied to other related problems.