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

Abstract

We discuss a poly-Laplacian system involving concave-convex nonlinearities and parameters subject to the Dirichlet boundary condition on locally finite graphs. It is obtained that the system admits at least one nontrivial solution of positive energy and one nontrivial solution of negative energy based on the mountain pass theorem and the Ekeland's variational principle. We also obtain an estimate about semi-trivial solutions. Moreover, by using a result due to Brown et al., which is based on the fibering method and the Nehari manifold, we get the existence of the ground-state solution to the single equation corresponding to the poly-Laplacian system. Especially, we present some ranges of parameters for all of the results.

Keywords

• poly-laplacian system
• concave-convex nonlinearities
• mountain pass theorem
• ekeland's variational principle
• locally finite graph