Results on homoclinic solutions of a partial difference equation involving the mean curvature operator

Yuhua Long; Sha Li; Yuhua Long; Sha Li

doi:10.3934/math.2025293

AIMS Mathematics

2025, Volume 10, Issue 3: 6429-6447. doi: 10.3934/math.2025293

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Research article

Results on homoclinic solutions of a partial difference equation involving the mean curvature operator

Yuhua Long ^{1,2
,
,},
Sha Li ^1,2

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

Received: 25 November 2024 Revised: 20 February 2025 Accepted: 18 March 2025 Published: 21 March 2025
MSC : Primary: 39A14; Secondary: 34C37

By variational technique coupled with the mountain pass lemma and fountain theorem, we investigate a second-order partial difference equation involving the mean curvature operator in the present paper. We establish a number of criteria to guarantee the existence of multiple nontrivial homoclinic solutions. Our results generalize and improve some known ones. Additionally, two examples are provided to demonstrate applications of our obtained results.
- variational technique,
- mountain pass lemma,
- fountain theorem,
- partial difference equation,
- homoclinic solution,
- mean curvature operator
Citation: Yuhua Long, Sha Li. Results on homoclinic solutions of a partial difference equation involving the mean curvature operator[J]. AIMS Mathematics, 2025, 10(3): 6429-6447. doi: 10.3934/math.2025293

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Abstract

By variational technique coupled with the mountain pass lemma and fountain theorem, we investigate a second-order partial difference equation involving the mean curvature operator in the present paper. We establish a number of criteria to guarantee the existence of multiple nontrivial homoclinic solutions. Our results generalize and improve some known ones. Additionally, two examples are provided to demonstrate applications of our obtained results.

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