# Existence of solutions for degenerated problems in <var>L</var><sup>1</sup> 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

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## 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 W01,p(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 L1(Ω).

## Cite this article

L. Aharouch, Elhoussine Azroul, A. Benkirane, Existence of solutions for degenerated problems in <var>L</var><sup>1</sup> having lower order terms with natural growth. Port. Math. 65 (2008), no. 1, pp. 95–120

DOI 10.4171/PM/1801