Existence of solutions for a poly-Laplacian system involving concave-convex nonlinearity on locally finite graphs

TL;DR

This paper proves the existence of multiple solutions for a poly-Laplacian system with nonlinearities on locally finite graphs using variational methods, providing parameter ranges and energy estimates.

Contribution

It introduces new existence results for solutions of a poly-Laplacian system on graphs, employing mountain pass, Ekeland's principle, and fibering methods.

Findings

Existence of at least two nontrivial solutions with different energies.

Parameter ranges where solutions exist are explicitly identified.

Ground state solutions for the associated single equation are established.

Abstract

We investigate the existence of two nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities and parameters with Dirichlet boundary condition on locally finite graphs. By using the mountain pass theorem and Ekeland's variational principle, we obtain that system has at least one nontrivial solution of positive energy and one nontrivial solution of negative energy, respectively. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result in [4] which is based on the fibering method and Nehari manifold, we obtain the existence of ground state solution to the single equation corresponding to poly-Laplacian system. Especially, we present some ranges of parameters in all of results.