Positive solutions for a semipositone anisotropic p-Laplacian problem
Abstract In this paper, a semipositone anisotropic p-Laplacian problem − Δ p → u = λ f ( u ) , $$ -\Delta _{\overrightarrow{p}}u=\lambda f(u), $$ on a bounded domain with the Dirchlet boundary condition is considered, where A ( u q − 1 ) ≤ f ( u ) ≤ B ( u q − 1 ) $A(u^{q}-1)\leq f(u)\leq B(u^{q}-1)$...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2024-03-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-024-01841-7 |
| _version_ | 1797273831213105152 |
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| author | A. Razani Giovany M. Figueiredo |
| author_facet | A. Razani Giovany M. Figueiredo |
| author_sort | A. Razani |
| collection | DOAJ |
| description | Abstract In this paper, a semipositone anisotropic p-Laplacian problem − Δ p → u = λ f ( u ) , $$ -\Delta _{\overrightarrow{p}}u=\lambda f(u), $$ on a bounded domain with the Dirchlet boundary condition is considered, where A ( u q − 1 ) ≤ f ( u ) ≤ B ( u q − 1 ) $A(u^{q}-1)\leq f(u)\leq B(u^{q}-1)$ for u > 0 $u>0$ , f ( 0 ) < 0 $f(0)<0$ and f ( u ) = 0 $f(u)=0$ for u ≤ − 1 $u\leq -1$ . It is proved that there exists λ ∗ > 0 $\lambda ^{*}>0$ such that if λ ∈ ( 0 , λ ∗ ) $\lambda \in (0,\lambda ^{*})$ , then the problem has a positive weak solution u λ ∈ L ∞ ( Ω ‾ ) $u_{\lambda}\in L^{\infty}(\overline{\Omega})$ via combining Mountain-Pass arguments, comparison principles, and regularity principles. |
| first_indexed | 2024-03-07T14:49:51Z |
| format | Article |
| id | doaj.art-b171515b57a74fd88537dfdb5555181e |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-03-07T14:49:51Z |
| publishDate | 2024-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-b171515b57a74fd88537dfdb5555181e2024-03-05T19:48:05ZengSpringerOpenBoundary Value Problems1687-27702024-03-012024111310.1186/s13661-024-01841-7Positive solutions for a semipositone anisotropic p-Laplacian problemA. Razani0Giovany M. Figueiredo1Department of Pure Mathematics, Faculty of Science, Imam Khomeini International UniversityDepartamento de Matemática, Universidade de BrasíliaAbstract In this paper, a semipositone anisotropic p-Laplacian problem − Δ p → u = λ f ( u ) , $$ -\Delta _{\overrightarrow{p}}u=\lambda f(u), $$ on a bounded domain with the Dirchlet boundary condition is considered, where A ( u q − 1 ) ≤ f ( u ) ≤ B ( u q − 1 ) $A(u^{q}-1)\leq f(u)\leq B(u^{q}-1)$ for u > 0 $u>0$ , f ( 0 ) < 0 $f(0)<0$ and f ( u ) = 0 $f(u)=0$ for u ≤ − 1 $u\leq -1$ . It is proved that there exists λ ∗ > 0 $\lambda ^{*}>0$ such that if λ ∈ ( 0 , λ ∗ ) $\lambda \in (0,\lambda ^{*})$ , then the problem has a positive weak solution u λ ∈ L ∞ ( Ω ‾ ) $u_{\lambda}\in L^{\infty}(\overline{\Omega})$ via combining Mountain-Pass arguments, comparison principles, and regularity principles.https://doi.org/10.1186/s13661-024-01841-7Mountain-Pass TheoremSemipositone problemAnisotropic p-LaplacianPositive solutions |
| spellingShingle | A. Razani Giovany M. Figueiredo Positive solutions for a semipositone anisotropic p-Laplacian problem Boundary Value Problems Mountain-Pass Theorem Semipositone problem Anisotropic p-Laplacian Positive solutions |
| title | Positive solutions for a semipositone anisotropic p-Laplacian problem |
| title_full | Positive solutions for a semipositone anisotropic p-Laplacian problem |
| title_fullStr | Positive solutions for a semipositone anisotropic p-Laplacian problem |
| title_full_unstemmed | Positive solutions for a semipositone anisotropic p-Laplacian problem |
| title_short | Positive solutions for a semipositone anisotropic p-Laplacian problem |
| title_sort | positive solutions for a semipositone anisotropic p laplacian problem |
| topic | Mountain-Pass Theorem Semipositone problem Anisotropic p-Laplacian Positive solutions |
| url | https://doi.org/10.1186/s13661-024-01841-7 |
| work_keys_str_mv | AT arazani positivesolutionsforasemipositoneanisotropicplaplacianproblem AT giovanymfigueiredo positivesolutionsforasemipositoneanisotropicplaplacianproblem |