Existence of solutions to Schrödinger systems on locally finite graphs

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Abstract

We consider the existence of solutions for some Schrödinger systems on locally finite graphs. Using variational 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.

Keywords

Schrödinger system / locally finite graphs / Lusternik-Schnirelmann theory / Nehari manifold / variational method

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