Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems

Yuhua Long; Huan Zhang; Yuhua Long; Huan Zhang

doi:10.3934/era.2022137

Electronic Research Archive

2022, Volume 30, Issue 7: 2681-2699. doi: 10.3934/era.2022137

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Research article

Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems

Yuhua Long ^{1,2
,
,},
Huan Zhang ^1,2

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
Guangzhou University, Guangzhou Center for Applied Mathematics, Guangzhou 510006, China

Academic Editor: Alain Miranville

Received: 15 April 2022 Revised: 06 May 2022 Accepted: 09 May 2022 Published: 18 May 2022

In this paper, we study discrete elliptic Dirichlet problems. Applying a variational technique together with Morse theory, we establish several results on the existence and multiplicity of nontrivial solutions. Finally, two examples and numerical simulations are provided to illustrate our theoretical results.
- discrete elliptic Dirichlet problem,
- local linking,
- Morse theory,
- nontrivial solution
Citation: Yuhua Long, Huan Zhang. Existence and multiplicity of nontrivial solutions to discrete elliptic Dirichlet problems[J]. Electronic Research Archive, 2022, 30(7): 2681-2699. doi: 10.3934/era.2022137

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Abstract

In this paper, we study discrete elliptic Dirichlet problems. Applying a variational technique together with Morse theory, we establish several results on the existence and multiplicity of nontrivial solutions. Finally, two examples and numerical simulations are provided to illustrate our theoretical results.

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