Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities
This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, Δp2u=uγ−1u+μu−1−α/xβu in Ω and u=∂u/∂n=0 on ∂Ω, where 4<2p<N,0∈Ω, −∞<μ<μ∗,=N−2p1−α/pN,p<γ<p∗=pN/N−2p is the cri...
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2023-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2023/7311332 |
| _version_ | 1826831437040451584 |
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| author | Mohammed El Mokhtar Ould El Mokhtar |
| author_facet | Mohammed El Mokhtar Ould El Mokhtar |
| author_sort | Mohammed El Mokhtar Ould El Mokhtar |
| collection | DOAJ |
| description | This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, Δp2u=uγ−1u+μu−1−α/xβu in Ω and u=∂u/∂n=0 on ∂Ω, where 4<2p<N,0∈Ω, −∞<μ<μ∗,=N−2p1−α/pN,p<γ<p∗=pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1. Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in E by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when Ω⊂ℝN be a bounded, smoothandstrictlystar-shapeddomain, β=0 and γ≥γ∗,=pN1−α/N−2p1−α−μNp>p∗=pN/N−2p. |
| first_indexed | 2024-04-09T23:33:04Z |
| format | Article |
| id | doaj.art-8756fac770bc40afb490039b7b71d07d |
| institution | Directory Open Access Journal |
| issn | 2314-8888 |
| language | English |
| last_indexed | 2025-02-16T09:59:41Z |
| publishDate | 2023-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj.art-8756fac770bc40afb490039b7b71d07d2025-02-03T01:30:44ZengWileyJournal of Function Spaces2314-88882023-01-01202310.1155/2023/7311332Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular NonlinearitiesMohammed El Mokhtar Ould El Mokhtar0Department of MathematicsThis article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, Δp2u=uγ−1u+μu−1−α/xβu in Ω and u=∂u/∂n=0 on ∂Ω, where 4<2p<N,0∈Ω, −∞<μ<μ∗,=N−2p1−α/pN,p<γ<p∗=pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1. Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in E by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when Ω⊂ℝN be a bounded, smoothandstrictlystar-shapeddomain, β=0 and γ≥γ∗,=pN1−α/N−2p1−α−μNp>p∗=pN/N−2p.http://dx.doi.org/10.1155/2023/7311332 |
| spellingShingle | Mohammed El Mokhtar Ould El Mokhtar Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities Journal of Function Spaces |
| title | Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities |
| title_full | Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities |
| title_fullStr | Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities |
| title_full_unstemmed | Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities |
| title_short | Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities |
| title_sort | existence and nonexistence for boundary problem involving the p biharmonic operator and singular nonlinearities |
| url | http://dx.doi.org/10.1155/2023/7311332 |
| work_keys_str_mv | AT mohammedelmokhtarouldelmokhtar existenceandnonexistenceforboundaryprobleminvolvingthepbiharmonicoperatorandsingularnonlinearities |