Mathematica Bohemica, Vol. 148, No. 2, pp. 255-282, 2023
Existence of renormalized solutions for some degenerate and non-coercive elliptic equations
Youssef Akdim, Mohammed Belayachi, Hassane Hjiaj
Received May 4, 2021. Published online June 23, 2022.
Abstract: This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by
$-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^p = f - {\rm div}(c(x)|u|^{\alpha})$ in $\Omega$,
$u = 0$ on $\partial\Omega$, where $\Omega$ is a bounded open set of $\mathbb{R}^N$ ($N\geq2$) with $1<p<N$ and $f \in L^1(\Omega),$ under some growth conditions on the function $b(\cdot)$ and $d(\cdot),$ where $c(\cdot)$ is assumed to be in $L^{\frc{N}{(p-1)}}(\Omega).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.
Keywords: renormalized solution; nonlinear elliptic equation; non-coercive problem
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Affiliations: Youssef Akdim, Mohammed Belayachi (corresponding author), Department of Mathematics, Laboratory LAMA, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, BP 1796 Atlas, Fez, Morocco, e-mail: youssef.akdim@usmba.ac.ma, mohammedbelayachi7@gmail.com; Hassane Hjiaj, Department of Mathematics, Faculty of Sciences Tetouan, University Abdelmalek Essaadi, Quartier M'haneche II, Avenue Palestine, BP 2121, Tetouan 93000, Morocco, e-mail: hjiajhassane@yahoo.fr
