Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition
We investigate the singular Neumann problem involving the p(x)-Laplace operator: Pλ{-Δpxu+|u|px-2u =1/uδx+fx,u, in Ω; u>0, in Ω; ∇upx-2∂u/∂ν=λuqx, on ∂Ω}, where Ω⊂RNN≥2 is a bounded domain with C2 boundary, λ is a positive parameter, and px,qx,δx, and fx,u are assumed to satisfy assumptio...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2016/3149482 |
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| author | K. Saoudi M. Kratou S. Alsadhan |
| author_facet | K. Saoudi M. Kratou S. Alsadhan |
| author_sort | K. Saoudi |
| collection | DOAJ |
| description | We investigate the singular Neumann problem involving the p(x)-Laplace operator: Pλ{-Δpxu+|u|px-2u =1/uδx+fx,u, in Ω; u>0, in Ω; ∇upx-2∂u/∂ν=λuqx, on ∂Ω}, where Ω⊂RNN≥2 is a bounded domain with C2 boundary, λ is a positive parameter, and px,qx,δx, and fx,u are assumed to satisfy assumptions (H0)–(H5) in the Introduction. Using some variational techniques, we show the existence of a number Λ∈0,∞ such that problem Pλ has two solutions for λ∈0,Λ, one solution for λ=Λ, and no solutions for λ>Λ. |
| format | Article |
| id | doaj-art-3c84eae6a3e144ccb9a2c502862aa21d |
| institution | Kabale University |
| issn | 1687-9643 1687-9651 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-3c84eae6a3e144ccb9a2c502862aa21d2025-02-03T05:58:45ZengWileyInternational Journal of Differential Equations1687-96431687-96512016-01-01201610.1155/2016/31494823149482Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary ConditionK. Saoudi0M. Kratou1S. Alsadhan2College of Sciences at Dammam, University of Dammam, Dammam 31441, Saudi ArabiaCollege of Sciences at Dammam, University of Dammam, Dammam 31441, Saudi ArabiaCollege of Sciences at Dammam, University of Dammam, Dammam 31441, Saudi ArabiaWe investigate the singular Neumann problem involving the p(x)-Laplace operator: Pλ{-Δpxu+|u|px-2u =1/uδx+fx,u, in Ω; u>0, in Ω; ∇upx-2∂u/∂ν=λuqx, on ∂Ω}, where Ω⊂RNN≥2 is a bounded domain with C2 boundary, λ is a positive parameter, and px,qx,δx, and fx,u are assumed to satisfy assumptions (H0)–(H5) in the Introduction. Using some variational techniques, we show the existence of a number Λ∈0,∞ such that problem Pλ has two solutions for λ∈0,Λ, one solution for λ=Λ, and no solutions for λ>Λ.http://dx.doi.org/10.1155/2016/3149482 |
| spellingShingle | K. Saoudi M. Kratou S. Alsadhan Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition International Journal of Differential Equations |
| title | Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition |
| title_full | Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition |
| title_fullStr | Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition |
| title_full_unstemmed | Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition |
| title_short | Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition |
| title_sort | multiplicity results for the px laplacian equation with singular nonlinearities and nonlinear neumann boundary condition |
| url | http://dx.doi.org/10.1155/2016/3149482 |
| work_keys_str_mv | AT ksaoudi multiplicityresultsforthepxlaplacianequationwithsingularnonlinearitiesandnonlinearneumannboundarycondition AT mkratou multiplicityresultsforthepxlaplacianequationwithsingularnonlinearitiesandnonlinearneumannboundarycondition AT salsadhan multiplicityresultsforthepxlaplacianequationwithsingularnonlinearitiesandnonlinearneumannboundarycondition |