Existence theory for elliptic equations of general exponential nonlinearity on finite graphs

2025/05/20 by Bobo Hua, Hua, Bobo, Linlin Sun +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2505.14799

openalex publication_date 2025/05/20 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

Abstract

We study semilinear elliptic equations on finite graphs with fully general exponential nonlinearities, thereby extending classical equations such as the Kazdan-Warner and Chern-Simons equations. A key contribution of this work is the development of new techniques for deriving a priori estimates in this generalized setting, which reduce the original finite graph to a graph with only two vertices. This reduction enables us to explicitly compute the Brouwer degree and to establish the existence of solutions when the degree is nonzero. Furthermore, using the method of sub- and supersolutions, we also prove the existence of solutions in cases where the Brouwer degree vanishes.

Citations

• On topological solutions to a generalized Chern-Simons equation on lattice graphs
• Sinh-Gordon equations on finite graphs
• Solutions to a generalized Chern-Simons Higgs model on finite graphs by topological degree
• The existence of topological solutions to the Chern-Simons model on lattice graphs
• Topological degree for Chern-Simons Higgs models on finite graphs
• Existence of Solutions to a Class of Kazdan-Warner Equations on Finite Graphs
• Existence of solutions to a generalized self-dual Chern-Simons system on finite graphs
• Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs
• Calculus of variations on locally finite graphs
• Existence of solutions to a generalized self-dual Chern-Simons equation on graphs
• Brouwer degree for Kazdan-Warner equations on a connected finite graph
• A note on Kazdan-Warner equation on networks
• A note on Liouville type equations on graphs
• The Kazdan-Warner equation on canonically compactifiable graphs
• Kazdan-Warner equation on infinite graphs
• Kazdan-Warner equation on graph

Related

• Existence results for Kazdan-Warner type equations on graphs
• Existence results for some nonlinear elliptic systems on graphs
• Nonexistence results for semilinear elliptic equations on weighted graphs
• Nonlinear elliptic Partial Differential Equations and p-harmonic functions on graphs
• Existence and convergence of solutions for nonlinear elliptic systems on graphs