Infinitely many solutions for a biharmonic-Kirchhoff system on locally finite graphs
Xiaoyu Wang, Junping Xie, Xingyong Zhang
TL;DR
This work analyzes a coupled biharmonic-Kirchhoff system on locally finite graphs, establishing the existence of infinitely many nontrivial solutions via variational methods and the symmetric mountain pass theorem. The authors formulate an energy functional on the product space $E=E_1\times E_2$ and verify the necessary compactness and geometric conditions under assumptions on the potentials $V_i$ and the nonlinearity $F$, with super-$4$-growth playing a key role. By exploiting the stronger embedding theorems available on locally finite graphs, they obtain a Palais-Smale framework and infinite sequences of critical points, improving upon corresponding Euclidean results. The paper also provides parallel results for the biharmonic system ($b_i=0$) and discusses the implications for scalar reductions, highlighting the discrete graph setting as advantageous for multiplicity results in high-order Kirchhoff-type problems.
Abstract
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.