# Existence of solutions for degenerated problems in $L_{1}$ having lower order terms with natural growth

### L. Aharouch

Université Ibn Zohr, Ouarzazate, Morocco### Elhoussine Azroul

Université Sidi Mohamed Ben Abdellah, Fès, Morocco### A. Benkirane

Université Sidi Mohamed Ben Abdellah, Fès, Morocco

## Abstract

We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation

$Au+g(x,u,∇u)=f,$

where $A$ is a Leray–Lions operator from the weighted Sobolev space $W_{0}(Ω, w)$ into its dual $W_{−1,p_{′}}(Ω, w_{∗})$. While $g(x,s,ξ)$ is a nonlinear term having natural growth with respect to $ξ$ and no growth with respect to $s$, it satisfies a sign condition on $s$, i.e., $g(x,s,ξ) ⋅ s≥0$ for every $s∈R$. The right-hand side $f$ belongs to $L_{1}(Ω)$.

## Cite this article

L. Aharouch, Elhoussine Azroul, A. Benkirane, Existence of solutions for degenerated problems in $L_{1}$ having lower order terms with natural growth. Port. Math. 65 (2008), no. 1, pp. 95–120

DOI 10.4171/PM/1801