On finite-energy solutions of Kazan-Warner equations on the lattice graph

2025/09/01 by Chen, Huyuan, hua, Bobo
#05C22 #35J91 #Analysis of PDEs (math.AP) #F.m #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.01155

Abstract

We investigate finite-energy solutions to Kazdan-Warner type equations in 2-dimensional integer lattice graph - Δu= ε e^κu +βδ₀ \rm in ℤ², where ε=±1, κ>0 and β∈ℝ. When ε=1, we prove the existence of a continuous family of finite-energy solutions for some parameter κ. This provides a partial resolution of the open problem on the existence of finite-energy solutions to the Liouville equation. When ε=-1 and β>\frac4πκ, we prove that the set of finite-energy solutions exhibits a layer structure. Moreover, we derive the extremal solution in this case.

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Related

• Existence results for Kazdan-Warner type equations on graphs
• Kazdan-Warner equation on infinite graphs
• A note on Liouville type equations on graphs
• p-th Kazdan-Warner equation on graph
• On the Lattice Potential KP Equation