Existence of Nontrivial Solutions for the Nonlinear Equation on Locally Finite Graphs

2023/10/11 by Ziliang Yang, Yang, Ziliang, Jiabao Su +3
Mathematics · #Advanced Mathematical Physics Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2310.07508

openalex publication_date 2023/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that G=(V, E) be a locally finite and connected graph with symmetric weight and uniformly positive measure, where V denotes the vertex set and E denotes the edge set. We are concered with the following problem \begincases-Δu+h u=f(x, u), amp; \text in Ω,
u=0, amp; \text on ∂ Ω,\endcases on the graph, where h: Ω→ ℝ, f: Ω× ℝ → ℝ and u: Ω→ ℝ. When f and h satisfies certain assumption conditions, we can ascertain the existence of one or two nontrivial solutions on the graph.

Related

• Existence and convergence of solutions to p-Laplace equations on locally finite graphs
• Blow-up problems for nonlinear parabolic equations on locally finite graphs
• Existence and multiplicity of nontrivial solutions for a (p,q)-Laplacian system on locally finite graphs
• Existence and convergence of solutions for nonlinear elliptic systems on graphs
• Superlinear elliptic inequalities on weighted graphs