Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Akdim, Youssef ; Belayachi, Mohammed ; Hjiaj, Hassane

Mathematica Bohemica, Tome 148 (2023) no. 2, pp. 255-282.

Résumé

This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by $$ \begin{aligned}t 2-{\rm div}( b(|u|)|\nabla u|^{p-2}\nabla u) + d(|u|)|\nabla u|^{p} = f - {\rm div}(c(x)|u|^{\alpha }) \quad \mbox {in}\ \Omega ,\\ u = 0 \quad \mbox {on}\ \partial \Omega , \end{aligned}t $$ where $\Omega $ is a bounded open set of $\mathbb {R}^N$ ($N\geq 2$) with $1$ 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 $L^{\frac {N}{(p-1)}}(\Omega ).$ We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.

Détail

DOI : 10.21136/MB.2022.0061-21

Classification : 35J60, 46E30, 46E35
Keywords: renormalized solution; nonlinear elliptic equation; non-coercive problem