On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces
In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations − div𝒜x,u,Δu ω1+ℬx,u,∇uν1+ℋx,u,∇uν2+up−2 u ω2−∑i,j=1nDjaijxDiux=f0x−∑j=1nDjfjx in Ω,ux=0 on ∂Ω,\begin{array}{*{20}{c}}{ - \;{\rm{div}}\left[ {\...
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2024-11-01
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| Series: | Annales Mathematicae Silesianae |
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| Online Access: | https://doi.org/10.2478/amsil-2024-0024 |
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| author | Cavalheiro Albo Carlos |
| author_facet | Cavalheiro Albo Carlos |
| author_sort | Cavalheiro Albo Carlos |
| collection | DOAJ |
| description | In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations
− div𝒜x,u,Δu ω1+ℬx,u,∇uν1+ℋx,u,∇uν2+up−2 u ω2−∑i,j=1nDjaijxDiux=f0x−∑j=1nDjfjx in Ω,ux=0 on ∂Ω,\begin{array}{*{20}{c}}{ - \;{\rm{div}}\left[ {\mathcal{A}\left( {x,u,\Delta u} \right)\;{\omega _1} + {\mathcal{B}}\left( {x,u,\nabla u} \right){\nu _1}} \right] + \mathcal{H}\left( {x,u,\nabla u} \right){\nu _2} + {{\left| u \right|}^{p - 2}}u\;{\omega _2}}\\{ - \sum\limits_{i,j = 1}^n {{D_j}\left( {{a_{ij}}\left( x \right){D_i}u\left( x \right)} \right)} = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)} \;\;\;\;\;{\rm{in}}\;\;\Omega ,}\\{u\left( x \right) = 0\;\;\;\;\;{\rm{on}}\;\partial \Omega ,}\end{array}
in the setting of the weighted Sobolev spaces. |
| format | Article |
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| institution | DOAJ |
| issn | 2391-4238 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Sciendo |
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| series | Annales Mathematicae Silesianae |
| spelling | doaj-art-32cd206baf614f96bd11cd3a1dd59e862025-08-20T02:41:11ZengSciendoAnnales Mathematicae Silesianae2391-42382024-11-0139222324710.2478/amsil-2024-0024On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev SpacesCavalheiro Albo Carlos0Department of Mathematics, State University of Londrina, Londrina, BrazilIn this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations − div𝒜x,u,Δu ω1+ℬx,u,∇uν1+ℋx,u,∇uν2+up−2 u ω2−∑i,j=1nDjaijxDiux=f0x−∑j=1nDjfjx in Ω,ux=0 on ∂Ω,\begin{array}{*{20}{c}}{ - \;{\rm{div}}\left[ {\mathcal{A}\left( {x,u,\Delta u} \right)\;{\omega _1} + {\mathcal{B}}\left( {x,u,\nabla u} \right){\nu _1}} \right] + \mathcal{H}\left( {x,u,\nabla u} \right){\nu _2} + {{\left| u \right|}^{p - 2}}u\;{\omega _2}}\\{ - \sum\limits_{i,j = 1}^n {{D_j}\left( {{a_{ij}}\left( x \right){D_i}u\left( x \right)} \right)} = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)} \;\;\;\;\;{\rm{in}}\;\;\Omega ,}\\{u\left( x \right) = 0\;\;\;\;\;{\rm{on}}\;\partial \Omega ,}\end{array} in the setting of the weighted Sobolev spaces.https://doi.org/10.2478/amsil-2024-002435j7035j6035j30degenerate nonlinear elliptic equationsweighted sobolev spaces |
| spellingShingle | Cavalheiro Albo Carlos On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces Annales Mathematicae Silesianae 35j70 35j60 35j30 degenerate nonlinear elliptic equations weighted sobolev spaces |
| title | On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces |
| title_full | On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces |
| title_fullStr | On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces |
| title_full_unstemmed | On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces |
| title_short | On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces |
| title_sort | on the dirichlet problem for a class of nonlinear degenerate elliptic equations in weighted sobolev spaces |
| topic | 35j70 35j60 35j30 degenerate nonlinear elliptic equations weighted sobolev spaces |
| url | https://doi.org/10.2478/amsil-2024-0024 |
| work_keys_str_mv | AT cavalheiroalbocarlos onthedirichletproblemforaclassofnonlineardegenerateellipticequationsinweightedsobolevspaces |