## Abstract

**We consider the nonlinear elliptic system** { u ∈ W 0 N N − 1 (
Ω ): − div ( M ( x ) ∇ u )+ u = − div ( u M ( x ) ∇ ψ )+ f ( x ) , ψ ∈ W
0 1 , 2 ( Ω ): − div ( M ( x ) ∇ ψ )+ ψ = R ( u )+ E ( x ) ∇ ψ ,
**where** Ω **is a bounded, open subset of** R N **,**
*N*≥3 **;** *M*( *x*) **is a coercive,
symmetric matrix with** L ∞ ( Ω ) **coefficients;** *f*(
*x*) **and** *E*( *x*) **belong to some
Lebesgue space, and** *R*( *s*) **is a continuous
function such that** 0 ≤ R ( s ) ≤ | s | θ , for θ
< 2 N . **Using a duality technique, we prove existence
of at least a weak solution** ( *u,ψ*) **. Moreover, if**
*N*=3 **or** *N*=4 **, we prove under stronger
assumptions on** *f*( *x*) **and** *E*( *x*)
**that the solution** *u* **belongs to** W 0 1 , 2 ( Ω )
**.**