Nontrivial solutions for a $(p,q)$-Kirchhoff type system with concave-convex nonlinearities on locally finite graphs

TL;DR

This paper establishes the existence of multiple solutions for a nonlinear Kirchhoff system on finite graphs using variational methods, and also discusses conditions for semi-trivial solutions and nonexistence.

Contribution

It introduces new existence and nonexistence results for a $(p,q)$-Kirchhoff system on graphs using variational techniques, including the mountain pass theorem and Ekeland's principle.

Findings

At least two fully-non-trivial solutions exist.

Conditions for semi-trivial solutions are characterized.

Nonexistence of solutions under certain conditions.

Abstract

By using the well-known mountain pass theorem and Ekeland's variational principle, we prove that there exist at least two fully-non-trivial 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 semi-trivial 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 semi-trivial solutions when the perturbation terms satisfy different assumptions. Finally, we present a nonexistence result of solutions.