On a nonlinear PDE involving weighted $p$-Laplacian
In the present paper, we study the nonlinear partial differential equation with the weighted $p$-Laplacian operator \begin{gather*} - \operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u) = \frac{ f(x)}{(1-u)^{2}}, \end{gather*} on a ball ${B}_{r}\subset \mathbb{R}^{N}(N\geq 2)$. Under some appropriate c...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Sociedade Brasileira de Matemática
2019-03-01
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| Series: | Boletim da Sociedade Paranaense de Matemática |
| Subjects: | |
| Online Access: | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/33978 |
| _version_ | 1797633473592164352 |
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| author | A. El Khalil M. D. Morchid Alaoui Mohamed Laghzal A. Touzani |
| author_facet | A. El Khalil M. D. Morchid Alaoui Mohamed Laghzal A. Touzani |
| author_sort | A. El Khalil |
| collection | DOAJ |
| description | In the present paper, we study the nonlinear partial differential equation with the weighted $p$-Laplacian operator
\begin{gather*}
- \operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u) = \frac{ f(x)}{(1-u)^{2}},
\end{gather*}
on a ball ${B}_{r}\subset \mathbb{R}^{N}(N\geq 2)$. Under some appropriate conditions
on the functions $f, w$ and the nonlinearity $\frac{1}{(1-u)^{2}}$, we prove the existence and the uniqueness of solutions of the above problem. Our analysis mainly combines the variational method and critical point theory. Such solution is obtained as a minimizer for the energy functional associated with our problem in the setting of the weighted Sobolev spaces. |
| first_indexed | 2024-03-11T11:54:30Z |
| format | Article |
| id | doaj.art-1bfb9926b4bc4132be50f654502ee241 |
| institution | Directory Open Access Journal |
| issn | 0037-8712 2175-1188 |
| language | English |
| last_indexed | 2024-03-11T11:54:30Z |
| publishDate | 2019-03-01 |
| publisher | Sociedade Brasileira de Matemática |
| record_format | Article |
| series | Boletim da Sociedade Paranaense de Matemática |
| spelling | doaj.art-1bfb9926b4bc4132be50f654502ee2412023-11-08T20:07:29ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882019-03-0138510.5269/bspm.v38i5.33978On a nonlinear PDE involving weighted $p$-LaplacianA. El Khalil0M. D. Morchid Alaoui1Mohamed Laghzal2A. Touzani3Al-Imam Mohammad Ibn Saud Islamic University (IMSIU) Department of Mathematics and StatisticsUniversity Sidi Mohamed Ben Abdellah Laboratory LAMA, Department of MathematicsUniversity Sidi Mohamed Ben Abdellah Laboratory LAMA, Department of MathematicsUniversity Sidi Mohamed Ben Abdellah Laboratory LAMA, Department of MathematicsIn the present paper, we study the nonlinear partial differential equation with the weighted $p$-Laplacian operator \begin{gather*} - \operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u) = \frac{ f(x)}{(1-u)^{2}}, \end{gather*} on a ball ${B}_{r}\subset \mathbb{R}^{N}(N\geq 2)$. Under some appropriate conditions on the functions $f, w$ and the nonlinearity $\frac{1}{(1-u)^{2}}$, we prove the existence and the uniqueness of solutions of the above problem. Our analysis mainly combines the variational method and critical point theory. Such solution is obtained as a minimizer for the energy functional associated with our problem in the setting of the weighted Sobolev spaces.https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/33978weighted $p$-Laplacian operatorSobolev spacesMuckenhoupt classexistenceuniqueness of solutions |
| spellingShingle | A. El Khalil M. D. Morchid Alaoui Mohamed Laghzal A. Touzani On a nonlinear PDE involving weighted $p$-Laplacian Boletim da Sociedade Paranaense de Matemática weighted $p$-Laplacian operator Sobolev spaces Muckenhoupt class existence uniqueness of solutions |
| title | On a nonlinear PDE involving weighted $p$-Laplacian |
| title_full | On a nonlinear PDE involving weighted $p$-Laplacian |
| title_fullStr | On a nonlinear PDE involving weighted $p$-Laplacian |
| title_full_unstemmed | On a nonlinear PDE involving weighted $p$-Laplacian |
| title_short | On a nonlinear PDE involving weighted $p$-Laplacian |
| title_sort | on a nonlinear pde involving weighted p laplacian |
| topic | weighted $p$-Laplacian operator Sobolev spaces Muckenhoupt class existence uniqueness of solutions |
| url | https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/33978 |
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