Existence and multiplicity of solutions for a class of $(p,q)$-Kirchhoff system with combined nonlinearities on graphs

By using the well-known mountain-pass theorem and Ekeland’s variational principle, we prove that there exist at least two fully nontrivial solutions for a $(p,q)$ -Kirchhoff elliptic system with the Dirichlet boundary conditions and perturbation terms on a locally weighted and connected finite graph $G=(V,E)$ . We also present a necessary condition of the existence of semitrivial solutions for the system. Moreover, by using Ekeland’s variational principle and Clark’s Theorem, respectively, we prove that the system has at least one or multiple semitrivial solutions when the perturbation terms satisfy different assumptions. Finally, we present a nonexistence result of solutions.