Existence of solutions to Schrödinger systems on locally finite graphs

Yan-Hong Chen; Hua Zhang; Yan-Hong Chen; Hua Zhang

doi:10.3934/era.2025329

Electronic Research Archive

2025, Volume 33, Issue 12: 7463-7490. doi: 10.3934/era.2025329

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Research article

Existence of solutions to Schrödinger systems on locally finite graphs

Yan-Hong Chen ^,,
Hua Zhang

Department of Mathematics, College of Information Science and Technology, Jinan University, Guangzhou 510632, China

Received: 12 August 2025 Revised: 16 November 2025 Accepted: 25 November 2025 Published: 10 December 2025

We consider the existence of solutions for some Schrödinger systems on locally finite graphs. Using variationial method and Nehari manifold method, the existence and multiplicity of nontrivial solutions are proved. More explicitly, we first give a functional setting of problem and prove the compactness of Sobolev embedding. Then for the Schrödinger system with quadratical nonlinear terms, the existence of nontrivial solutions is proved by using the Nehari manifold method. For the Schrödinger system with cubic type nonlinearity, we first prove the existence of ground state solution and then prove the multiplicity of solutions by combining the Nehari manifold method as well as the Lusternik-Schnirelmann theory.
- Schrödinger system,
- locally finite graphs,
- Lusternik-Schnirelmann theory,
- Nehari manifold,
- variational method
Citation: Yan-Hong Chen, Hua Zhang. Existence of solutions to Schrödinger systems on locally finite graphs[J]. Electronic Research Archive, 2025, 33(12): 7463-7490. doi: 10.3934/era.2025329

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Abstract

We consider the existence of solutions for some Schrödinger systems on locally finite graphs. Using variationial method and Nehari manifold method, the existence and multiplicity of nontrivial solutions are proved. More explicitly, we first give a functional setting of problem and prove the compactness of Sobolev embedding. Then for the Schrödinger system with quadratical nonlinear terms, the existence of nontrivial solutions is proved by using the Nehari manifold method. For the Schrödinger system with cubic type nonlinearity, we first prove the existence of ground state solution and then prove the multiplicity of solutions by combining the Nehari manifold method as well as the Lusternik-Schnirelmann theory.

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