Existence of entropy solutions of the anisotropic elliptic nonlinear problem with measure data in weighted Sobolev space

This paper is devoted to study the following nonlinear anisotropic elliptic unilateral problem \begin{equation*} \begin{cases} A\,u -\mbox{div}\,\phi(u)=\mu \quad \mbox{in} \qquad \Omega \\ \;u=0 \qquad \mbox{on} \quad \partial \Omega , \end{cases} \end{equation*} where the right hand side $\,\mu\;...

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Bibliographic Details
Main Authors: Adil Abbassi, Chakir Allalou, Abderrazak Kassidi
Format: Article
Language:English
Published: Sociedade Brasileira de Matemática 2022-02-01
Series:Boletim da Sociedade Paranaense de Matemática
Online Access:https://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/52541
Description
Summary:This paper is devoted to study the following nonlinear anisotropic elliptic unilateral problem \begin{equation*} \begin{cases} A\,u -\mbox{div}\,\phi(u)=\mu \quad \mbox{in} \qquad \Omega \\ \;u=0 \qquad \mbox{on} \quad \partial \Omega , \end{cases} \end{equation*} where the right hand side $\,\mu\;$ belongs to $\; L^1(\Omega)+ W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$. The operator $\displaystyle A\,u=-\sum_{i=1}^{N}\partial_{i}\,a_{i}(x,\ u,\ \nabla u)$ is a Leray-Lions anisotropic operator acting from $\; W_{0}^{1,\overrightarrow{p}} (\Omega,\ \overrightarrow{\omega})\;$ into its dual $\; W_{0}^{-1,\overrightarrow{p}'} (\Omega,\ \overrightarrow{\omega}^*)$ and $\phi_{i}\in C^{0}(\mathbb{R},\mathbb{R})$.
ISSN:0037-8712
2175-1188