标题： 无穷多解的双调和-Kirchhoff系统在局部有限图上的研究 显示英文标题

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

摘要： 关于图框架下偏微分方程（组）的研究近年来是一个热点话题，因为它们在图像处理和数据聚类中的应用。 我们的动机是将欧几里得框架下的双调和 - Kirchhoff 系统和双调和系统的某些存在性结果（这些是连续模型），推广到相应的局部有限图框架下的系统（这些是离散模型）。 我们主要关注局部有限图上双调和 - Kirchhoff 系统的无穷多解的存在性。 方法是变分法，主要工具是对称山路定理。 当非线性项具有超 -$4$ 线性增长时，我们得到该系统有无穷多个解，并且我们还给出了双调和系统的相应结果。 我们还发现局部有限图框架下的结果比欧几里得框架下的结果更好，这是由于局部有限图中有更好的嵌入定理。 显示英文摘要

摘要： 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.