The existence of solutions for nonlinear partial differential equations on locally finite graphs

In this paper, the variational method from local to global is used to generate results of second-order equations to higher-order equations, the existence of solutions for the higher-order Schrödinger equation and Kazdan-Warner equation on locally finite graphs is proved． The existence of rearrangement solutions for a class of nonlinear partial differential equations is proved using the Nehari manifold and rearrangement theory on higher-dimensional lattice graphs． Results of constrained variational problems are generalized to unconstrained variational problems．