A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term
Abstract Here, we consider the following elliptic problem with variable components: − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ with Dirichlet boundary...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-09-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-021-01557-y |
| _version_ | 1819153869983186944 |
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| author | MirKeysaan Mahshid Abdolrahman Razani |
| author_facet | MirKeysaan Mahshid Abdolrahman Razani |
| author_sort | MirKeysaan Mahshid |
| collection | DOAJ |
| description | Abstract Here, we consider the following elliptic problem with variable components: − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ with Dirichlet boundary condition in a bounded domain in R N $\mathbb{R}^{N}$ with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem. |
| first_indexed | 2024-12-22T15:12:03Z |
| format | Article |
| id | doaj.art-9300dae60d1c4bd99685f10a6016cd9c |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-22T15:12:03Z |
| publishDate | 2021-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-9300dae60d1c4bd99685f10a6016cd9c2022-12-21T18:21:51ZengSpringerOpenBoundary Value Problems1687-27702021-09-01202111910.1186/s13661-021-01557-yA weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular termMirKeysaan Mahshid0Abdolrahman Razani1Department of Mathematics, Islamic Azad UniversityDepartment of Pure Mathematics, Faculty of Science, Imam Khomeini International UniversityAbstract Here, we consider the following elliptic problem with variable components: − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ with Dirichlet boundary condition in a bounded domain in R N $\mathbb{R}^{N}$ with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.https://doi.org/10.1186/s13661-021-01557-y( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian problemSingular termVariational method |
| spellingShingle | MirKeysaan Mahshid Abdolrahman Razani A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term Boundary Value Problems ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian problem Singular term Variational method |
| title | A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term |
| title_full | A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term |
| title_fullStr | A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term |
| title_full_unstemmed | A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term |
| title_short | A weak solution for a ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian elliptic problem with a singular term |
| title_sort | weak solution for a p x q x p x q x laplacian elliptic problem with a singular term |
| topic | ( p ( x ) , q ( x ) ) $(p(x),q(x))$ -Laplacian problem Singular term Variational method |
| url | https://doi.org/10.1186/s13661-021-01557-y |
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