# Some existence results for a quasilinear problem with source term in Zygmund-space

### Boussad Hamour

Ecole Normale Supérieure, Alger, Algeria

## Abstract

In this paper we study the existence of solution to the problem

${u∈H_{0}(Ω),−div(A(x)Du)=H(x,u,Du)+f(x)+a_{0}(x)uinD_{′}(Ω), $

where $Ω$ is an open bounded set of $R_{2}$, $A(x)$ a coercive matrix with coefficients in $L_{∞}(Ω)$, $H(x,s,ξ)$ a Carathéodory function satisfying, for some $γ>0$,

$−c_{0}A(x)ξξ≤H(x,s,ξ)sign(s)≤γA(x)ξξa.e.x∈Ω,∀s∈R,∀ξ∈R_{2}.$

Here $f$ belongs to $L_{1}(gL_{1})(Ω)$ and $a_{0}≥0$ to $L_{q}(Ω)$, $q>1$. For $f$ and $a_{0}$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is such that $e_{δ_{0}∣u∣}−1$ belongs to $H_{0}(Ω)$ for some $δ_{0}≥γ$ and satisfies an *a priori* estimate.

## Cite this article

Boussad Hamour, Some existence results for a quasilinear problem with source term in Zygmund-space. Port. Math. 76 (2019), no. 3/4, pp. 259–286

DOI 10.4171/PM/2035