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On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces Cover

On the Dirichlet Problem for a Class of Nonlinear Degenerate Elliptic Equations in Weighted Sobolev Spaces

Open Access
|Nov 2024

Abstract

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations div𝒜x,u,Δuω1+x,u,uν1+x,u,uν2+up2uω2i,j=1nDjaijxDiux=f0xj=1nDjfjxinΩ,ux=0onΩ, \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.

DOI: https://doi.org/10.2478/amsil-2024-0024 | Journal eISSN: 2391-4238 | Journal ISSN: 0860-2107
Language: English
Page range: 223 - 247
Submitted on: Mar 11, 2024
Accepted on: Oct 28, 2024
Published on: Nov 19, 2024
Published by: University of Silesia in Katowice, Institute of Mathematics
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:

© 2024 Albo Carlos Cavalheiro, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution 4.0 License.