Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

The existence of solutions for two types of nonlinear equations on locally 1 finite graphs

Document Type : Original Article

Author
Zidong Qiu
Department of Mathematics and Physics, Zibo Normal College, Shandong, 255188, China
10.30511/ttmp.2025.2072701.1064
Abstract
In this paper, we focus on connected locally finite graphs G = (V, E). First, we assume that there are two constants μ_0 and ω_0, which make the measure function and symmetric weight function satisfy μ(x) ≥ μ_0 ∀, x ∈ V, ω_{xy} ≥ ω_0, ∀xy ∈ E. Based on this assumption, we obtain two interesting embedding theorems on finite graphs: W_0^{1,2} (B_k)↪L^p (B_k), W^{1,2} (B_k)↪L^p (B_k). Although their inclusion relations are obvious on finite graphs, here we mainly give the control relations under the same control coefficient. Secondly, Δ is the Laplace operator on a general graph. Due to Lin and Yang (2022), using calculus of variations from local to global, we establish the existence of solutions to the exponential power type nonlinear Schrödinger equation, says −Δu + hu = f u e^{u^2} + g, x ∈ V, and the existence of solutions for fractional nonlinear mean field equations, says −Δu + hu = (g e^u) / (∫_V g e^u dμ) + f /(u + m), x ∈ V. When f, g and h satisfy some conditions, we prove the existence of non explicit solutions for the above two kinds of equations in a specific space.
Keywords
Schrodinger equation
mean field equation
Sobolev embedding theorems
variational method
Subjects
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Transactions in Theoretical and Mathematical Physics
Volume 2, Issue 4
Autumn 2025
Pages 204-218

  • Receive Date 25 September 2025
  • Accept Date 28 November 2025
  • First Publish Date 28 November 2025
  • Publish Date 01 November 2025