Renormalized Solutions for some Nonlinear Degenerated Elliptic Problems

Document Type : Original Article

Authors

Mohammed El Fatry ¹

Mounir Mekkour ²

Youssef Akdim ²

¹ USMBA FESFaculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Laboratory L2MASI, B.P. 1796, Atlas Fez, Morocco.

² Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University, Laboratory L2MASI, B.P. 1796, Atlas Fez, Morocco.

10.22034/kjm.2025.506507.3462

Abstract

‎In this paper‎, ‎we prove the existence of renormalized solution for a class of nonlinear degenerated elliptic problem in the form
\begin{equation*}‎
‎ \left\lbrace \begin{aligned}‎ -‎\sum\limits_{i=1}^{N}\frac{\partial }{\partial‎
‎ x_{i}}\left( \sum\limits_{j=1}^{N}a_{ij}(x,u)\frac{\partial u}{\partial‎
‎ x_{j}}\right) \omega _{i}(x) &=f & & \text{in}\ \Omega‎, ‎\\ u &=0 & & \text‎{on}\ \partial \Omega‎, ‎\end{aligned}\right.
‎\end{equation*}
where $A(x,u)=\left(‎a_{ij}(x,s)\right) _{ 1\leq i,j\leq N}$ is blow-up for a finite value $m$ of the unknown $u$ and the data $ f\in L^{1}(\Omega) $‎.
 

Keywords

Weighted Sobolev spaces

renormalized solutions

Blow-up

$L^{1}$-data

Subjects

• 35 Partial differential equations

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