Infinitely many solutions for a biharmonic-Kirchhoff system on locally finite graphs

The study on the partial differential equations (systems) in the graph setting is a hot topic in recent years because of their applications to image processing and data clustering. Our motivation is to develop some existence results for biharmonic-Kirchhoff systems and biharmonic systems in the Euclidean setting, which are the continuous models, to the corresponding systems in the locally finite graph setting, which are the discrete models. We mainly focus on the existence of infinitely many solutions for a biharmonic-Kirchhoff system on a locally finite graph. The method is variational and the main tool is the symmetric mountain pass theorem. We obtain that the system has infinitely many solutions when the nonlinear term admits the super-$4$ linear growth, and we also present the corresponding results to the biharmonic system. We also find that the results in the locally finite graph setting are better than that in the Euclidean setting, which caused by the better embedding theorem in the locally finite graph.