Document Type : Research Paper

**Authors**

^{1}
Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou 510440, China.

^{2}
Oriental Science and Technology College, Hunan Agricultural University, Changsha 410128, China, and Science College, Hunan Agricultural University, Changsha 410128, China.

^{3}
Packaging Engineering Institute, Jinan University, Zhuhai 519070, China.

**Abstract**

This paper is concerned with a 2nth-order p-Laplacian difference equation. By using the

critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Neumann boundary value problem and give some new results. Results obtained successfully generalize and complement the existing ones.

critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Neumann boundary value problem and give some new results. Results obtained successfully generalize and complement the existing ones.

**Keywords**

- Nonexistence and existence
- Neumann boundary value problem
- 2$n$th-order $p$-Laplacian
- Mountain Pass lemma
- discrete variational theory

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December 2016

Pages 1507-1521

**Receive Date:**15 November 2013**Revise Date:**13 February 2015**Accept Date:**24 September 2015