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

TL;DR

The paper studies a nonhomogeneous $(p,q)$-Kirchhoff elliptic system with Dirichlet boundary conditions on a bounded domain within a locally finite graph, featuring concave-convex nonlinearities and perturbations. Using a variational framework on $X=W_0^{1,p}(\Omega)\times W_0^{1,q}(\Omega)$ and critical-point theory (Mountain Pass Theorem, Ekeland's variational principle, and Clark's Theorem), it proves the existence of at least two fully-non-trivial solutions (one with positive energy and one with negative energy) under hypotheses $(H_1)-(H_4)$. The work also provides a necessary condition for semi-trivial solutions, establishes the existence or multiplicity of semi-trivial solutions under different perturbation assumptions, and presents a nonexistence result under a global negativity condition. Overall, it extends variational methods for Kirchhoff-type systems to discrete graph settings, enriching the theory of nonlinear elliptic problems on graphs and guiding further study of nonlocal and nonhomogeneous terms in discrete media.

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.