Steklov problems for the p−Laplace operator involving Lq-norm
In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form {Δpu=|u|p-2uin Ω,|∇u|p-2∂u∂v=λ‖u‖q,∂Ωp-q|u|q-2uon ∂Ω,\left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Sciendo
2022-05-01
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| Series: | Moroccan Journal of Pure and Applied Analysis |
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| Online Access: | https://doi.org/10.2478/mjpaa-2022-0016 |
| _version_ | 1811254959461629952 |
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| author | Alaoui My Driss Morchid Khalil Abdelouahd El Touzani Abdelfattah |
| author_facet | Alaoui My Driss Morchid Khalil Abdelouahd El Touzani Abdelfattah |
| author_sort | Alaoui My Driss Morchid |
| collection | DOAJ |
| description | In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form
{Δpu=|u|p-2uin Ω,|∇u|p-2∂u∂v=λ‖u‖q,∂Ωp-q|u|q-2uon ∂Ω,\left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right.
where Ω is a smooth bounded domain in ℝN(N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W1,p(Ω). Using the Ljusterneck-Shnirelmann theory on C1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λk)k≥1, for the above problem. We then establish that the first eigenvalue is simple and isolated. |
| first_indexed | 2024-04-12T17:16:30Z |
| format | Article |
| id | doaj.art-0973dec4433941d0a286977955865bf7 |
| institution | Directory Open Access Journal |
| issn | 2351-8227 |
| language | English |
| last_indexed | 2024-04-12T17:16:30Z |
| publishDate | 2022-05-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Moroccan Journal of Pure and Applied Analysis |
| spelling | doaj.art-0973dec4433941d0a286977955865bf72022-12-22T03:23:38ZengSciendoMoroccan Journal of Pure and Applied Analysis2351-82272022-05-018222824310.2478/mjpaa-2022-0016Steklov problems for the p−Laplace operator involving Lq-normAlaoui My Driss Morchid0Khalil Abdelouahd El1Touzani Abdelfattah2Laboratory MAIS (AMNEA Group), Department of Mathematics, Faculty of Sciences and Technologies Moulay Ismail University of Meknes, BP 509, Boutalamine, 52000 Errachidia, MoroccoDepartment of Mathematics and Statistics, College of Science Al Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950, 11623Riyadh, KSALaboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz University Sidi Mohamed Ben Abdellah, P.O. Box 1796 Atlas Fez, MoroccoIn this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form {Δpu=|u|p-2uin Ω,|∇u|p-2∂u∂v=λ‖u‖q,∂Ωp-q|u|q-2uon ∂Ω,\left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right. where Ω is a smooth bounded domain in ℝN(N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W1,p(Ω). Using the Ljusterneck-Shnirelmann theory on C1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λk)k≥1, for the above problem. We then establish that the first eigenvalue is simple and isolated.https://doi.org/10.2478/mjpaa-2022-0016p-laplace operatornonlinear eigenvalue problemboundary value problemssobolev trace embeddingc1 manifoldljusternick-schnirelmann theory35j6658e05246e3558e05 |
| spellingShingle | Alaoui My Driss Morchid Khalil Abdelouahd El Touzani Abdelfattah Steklov problems for the p−Laplace operator involving Lq-norm Moroccan Journal of Pure and Applied Analysis p-laplace operator nonlinear eigenvalue problem boundary value problems sobolev trace embedding c1 manifold ljusternick-schnirelmann theory 35j66 58e052 46e35 58e05 |
| title | Steklov problems for the p−Laplace operator involving Lq-norm |
| title_full | Steklov problems for the p−Laplace operator involving Lq-norm |
| title_fullStr | Steklov problems for the p−Laplace operator involving Lq-norm |
| title_full_unstemmed | Steklov problems for the p−Laplace operator involving Lq-norm |
| title_short | Steklov problems for the p−Laplace operator involving Lq-norm |
| title_sort | steklov problems for the p laplace operator involving lq norm |
| topic | p-laplace operator nonlinear eigenvalue problem boundary value problems sobolev trace embedding c1 manifold ljusternick-schnirelmann theory 35j66 58e052 46e35 58e05 |
| url | https://doi.org/10.2478/mjpaa-2022-0016 |
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