Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions
Given a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞), we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω. In the first part of the article, we give nece...
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Wiley
2017-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2017/5196513 |
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| author | Ciprian G. Gal Mahamadi Warma |
| author_facet | Ciprian G. Gal Mahamadi Warma |
| author_sort | Ciprian G. Gal |
| collection | DOAJ |
| description | Given a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞), we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω. In the first part of the article, we give necessary and sufficient conditions in terms of the given functions f, g and the nonlinearities α1, α2, for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L∞-setting. |
| format | Article |
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| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-0400c2e573594d79a910f901bdc3afd92025-02-03T05:53:16ZengWileyAdvances in Mathematical Physics1687-91201687-91392017-01-01201710.1155/2017/51965135196513Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary ConditionsCiprian G. Gal0Mahamadi Warma1Department of Mathematics, Florida International University, Miami, FL 33199, USADepartment of Mathematics, Faculty of Natural Sciences, University of Puerto Rico, Rio Piedras Campus, P.O. Box 70377, San Juan, PR 00936-8377, USAGiven a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω and p,q∈(1,+∞), we consider the quasilinear elliptic equation -Δpu+α1u=f in Ω complemented with the generalized Wentzell-Robin type boundary conditions of the form bx∇up-2∂nu-ρbxΔq,Γu+α2u=g on ∂Ω. In the first part of the article, we give necessary and sufficient conditions in terms of the given functions f, g and the nonlinearities α1, α2, for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an L∞-setting.http://dx.doi.org/10.1155/2017/5196513 |
| spellingShingle | Ciprian G. Gal Mahamadi Warma Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions Advances in Mathematical Physics |
| title | Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions |
| title_full | Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions |
| title_fullStr | Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions |
| title_full_unstemmed | Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions |
| title_short | Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions |
| title_sort | nonlinear elliptic boundary value problems at resonance with nonlinear wentzell boundary conditions |
| url | http://dx.doi.org/10.1155/2017/5196513 |
| work_keys_str_mv | AT ciprianggal nonlinearellipticboundaryvalueproblemsatresonancewithnonlinearwentzellboundaryconditions AT mahamadiwarma nonlinearellipticboundaryvalueproblemsatresonancewithnonlinearwentzellboundaryconditions |