On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian
In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ-Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associate...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2021/7013733 |
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| _version_ | 1832561393219403776 |
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| author | Zihua Qiu |
| author_facet | Zihua Qiu |
| author_sort | Zihua Qiu |
| collection | DOAJ |
| description | In this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ-Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results. |
| format | Article |
| id | doaj-art-e65beee479894690afa5d6aaea5eeded |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-e65beee479894690afa5d6aaea5eeded2025-02-03T01:25:09ZengWileyJournal of Function Spaces2314-88962314-88882021-01-01202110.1155/2021/70137337013733On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-LaplacianZihua Qiu0School of Mathematics and Information Science, Guangzhou University, Guangdong Guangzhou 510006, ChinaIn this paper, we consider the multiplicity of solutions for a discrete boundary value problem involving the singular ϕ-Laplacian. In order to apply the critical point theory, we extend the domain of the singular operator to the whole real numbers. Instead, we consider an auxiliary problem associated with the original one. We show that, if the nonlinear term oscillates suitably at the origin, there exists a sequence of pairwise distinct nontrivial solutions with the norms tend to zero. By our strong maximum principle, we show that all these solutions are positive under some assumptions. Moreover, the solutions of the auxiliary problem are solutions of the original one if the solutions are appropriately small. Lastly, we give an example to illustrate our main results.http://dx.doi.org/10.1155/2021/7013733 |
| spellingShingle | Zihua Qiu On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian Journal of Function Spaces |
| title | On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian |
| title_full | On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian |
| title_fullStr | On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian |
| title_full_unstemmed | On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian |
| title_short | On the Multiplicity of Solutions for the Discrete Boundary Problem Involving the Singular ϕ-Laplacian |
| title_sort | on the multiplicity of solutions for the discrete boundary problem involving the singular ϕ laplacian |
| url | http://dx.doi.org/10.1155/2021/7013733 |
| work_keys_str_mv | AT zihuaqiu onthemultiplicityofsolutionsforthediscreteboundaryprobleminvolvingthesingularphlaplacian |