Critical point theory to isotropic discrete boundary value problems on weighted finite graphs

Abstract

In this paper, we use the Linking theorem, Clark theorem and Fountain theorem to obtain the existence of solutions to discrete boundary value problem involving the p-Laplacian on simple, connected, undirected and weighted graphs. We obtain three solutions, infinitely many solutions or a number of solutions depending on a parameter $\mathit{\lambda }$ and the number of vertices. The eigenvalues of the nonlinear operator $-{\mathrm{\Delta }}_{p,\mathit{\omega }}+q$ are concerned in the study.

Keywords:

• Discrete p-Laplacian
• Dirichlet boundary value problems
• critical point theory
• multiple solutions
• Eigenvalue

AMS Subject Classifications:

• 34B15
• 35A15
• 34B45

Acknowledgements

The authors are really grateful to the referee and editor for the useful comments and valuable suggestions which greatly improve the original manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by Research Innovation Fund for College Students of Beijing University of Posts and Communications [grant number 1707003]; the second author is supported by Beijing University of Posts and Telecommunications teaching reform project [2017JY23].