Existence of three solutions for a poly-Laplacian system on graphs

TL;DR

This paper proves the existence of three solutions for certain poly-Laplacian systems on graphs using an abstract critical points theorem, overcoming specific analytical challenges related to the graph-based operators.

Contribution

It extends critical point theory to poly-Laplacian systems on graphs, addressing the difficulty of the Gâteaux derivative's invertibility due to the graph structure.

Findings

Established three solutions for poly-Laplacian systems on graphs.

Overcame analytical challenges related to the invertibility of the Gâteaux derivative.

Applied critical point theory to graph-based differential operators.

Abstract

We deal with the existence of three distinct solutions for a poly-Laplacian system with a parameter on finite graphs and a $(p,q)$-Laplacian system with a parameter on locally finite graphs.The main tool is an abstract critical points theorem in [Bonanno and Bisci, J.Math.Appl.Anal, 2011, 382(1): 1-8]. A key point in this paper is that we overcome the difficulty to prove that the G$\hat{\mbox{a}}$teaux derivative of the variational functional for poly-Laplacian operator admits a continuous inverse, which is caused by the special definition of the poly-Laplacian operator on graph and mutual coupling of two variables in system.