Positive solutions to the discrete boundary value problem involving the singular $ \phi $-Laplacian

Zhiqiang Huang; Zhan Zhou; Zhiqiang Huang; Zhan Zhou

doi:10.3934/math.2025798

AIMS Mathematics

2025, Volume 10, Issue 8: 17922-17939. doi: 10.3934/math.2025798

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Positive solutions to the discrete boundary value problem involving the singular $ \phi $-Laplacian

Zhiqiang Huang ¹,
Zhan Zhou ^{1,2
,
,}

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

Received: 06 July 2025 Revised: 28 July 2025 Accepted: 31 July 2025 Published: 07 August 2025
MSC : 39A10, 34B18, 39A13, 49J35

In this paper, we investigated the existence of solutions to the discrete boundary value problem involving the singular $ \phi $-Laplacian. First, we extended the domain of the singular operator to the entire real line, which leads to an auxiliary problem corresponding to the original one. Then, by using critical point theory combined with the strong maximum principle, we obtained a series of conditions for the existence of one positive solution or multiple positive solutions for the original problem. Finally, three numerical examples were provided to illustrate the effectiveness of our results.
- singular $ \phi $-Laplacian,
- boundary value problem,
- critical point theory,
- strong maximum principle,
- positive solutions
Citation: Zhiqiang Huang, Zhan Zhou. Positive solutions to the discrete boundary value problem involving the singular $ \phi $-Laplacian[J]. AIMS Mathematics, 2025, 10(8): 17922-17939. doi: 10.3934/math.2025798

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Abstract

In this paper, we investigated the existence of solutions to the discrete boundary value problem involving the singular $ \phi $-Laplacian. First, we extended the domain of the singular operator to the entire real line, which leads to an auxiliary problem corresponding to the original one. Then, by using critical point theory combined with the strong maximum principle, we obtained a series of conditions for the existence of one positive solution or multiple positive solutions for the original problem. Finally, three numerical examples were provided to illustrate the effectiveness of our results.

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