Brezis Nirenberg type results for local non-local problems under mixed boundary conditions
In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal const...
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| Format: | Article |
| Language: | English |
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AIMS Press
2024-11-01
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| Series: | Communications in Analysis and Mechanics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2024038 |
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| author | Lovelesh Sharma |
| author_facet | Lovelesh Sharma |
| author_sort | Lovelesh Sharma |
| collection | DOAJ |
| description | In this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases. |
| format | Article |
| id | doaj-art-8bedd2791aa34caabc9fe3ed0046f861 |
| institution | Kabale University |
| issn | 2836-3310 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj-art-8bedd2791aa34caabc9fe3ed0046f8612025-01-23T07:55:55ZengAIMS PressCommunications in Analysis and Mechanics2836-33102024-11-0116487289510.3934/cam.2024038Brezis Nirenberg type results for local non-local problems under mixed boundary conditionsLovelesh Sharma0Department of Mathematics, Indian Institute of Technology Jodhpur, Rajasthan 342030, IndiaIn this paper, we are concerned with an elliptic problem with mixed Dirichlet and Neumann boundary conditions that involve a mixed operator (i.e., the combination of classical Laplace operator and fractional Laplace operator) and critical nonlinearity. Also, we focus on identifying the optimal constant in the mixed Sobolev inequality, which we show is never achieved. Furthermore, by using variational methods, we provide an existence and nonexistence theory for both linear and superlinear perturbation cases.https://www.aimspress.com/article/doi/10.3934/cam.2024038mixed local-nonlocal operatorsmixed type sobolev inequalitymixed boundary conditionsexistence and non-existence resultsvariational methods |
| spellingShingle | Lovelesh Sharma Brezis Nirenberg type results for local non-local problems under mixed boundary conditions Communications in Analysis and Mechanics mixed local-nonlocal operators mixed type sobolev inequality mixed boundary conditions existence and non-existence results variational methods |
| title | Brezis Nirenberg type results for local non-local problems under mixed boundary conditions |
| title_full | Brezis Nirenberg type results for local non-local problems under mixed boundary conditions |
| title_fullStr | Brezis Nirenberg type results for local non-local problems under mixed boundary conditions |
| title_full_unstemmed | Brezis Nirenberg type results for local non-local problems under mixed boundary conditions |
| title_short | Brezis Nirenberg type results for local non-local problems under mixed boundary conditions |
| title_sort | brezis nirenberg type results for local non local problems under mixed boundary conditions |
| topic | mixed local-nonlocal operators mixed type sobolev inequality mixed boundary conditions existence and non-existence results variational methods |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2024038 |
| work_keys_str_mv | AT loveleshsharma brezisnirenbergtyperesultsforlocalnonlocalproblemsundermixedboundaryconditions |