Existence of three solutions for two quasilinear Laplacian systems on graphs

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)\left(p,q)-Laplacian system with a parameter on locally finite graphs. The main tool is an abstract critical point theorem in [G. Bonanno and S. A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), no. 1, 1–10]. A key point in this study is that we overcome the difficulty to prove that the Gâteaux 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.

Keywords

• critical point theorem
• ploy-laplacian system
• finite graph
• locally finite graph
• nontrivial solution
• 35j60
• 35j62
• 49j35