### Existence of weak solution for nonlinear elliptic system

#### Abstract

In this paper, we prove the existence of weak solutions for the following nonlinear elliptic system

*-*div *A*(*x,**∇**u*) = *-**a*(*x*)*|**u**|*^{p}^{(}^{x}^{)}^{-}^{2}*u **- **b*(*x*)*|**u**|*^{α}^{(}^{x}^{)}*|**v**|*^{β}^{(}^{x}^{)}*v *+ *f*(*x*) *in *Ω*,*

*-*div *B*(*x,**∇**v*) = *-**c*(*x*)*|**v**|*^{q}^{(}^{x}^{)}^{-}^{2}*v **- **d*(*x*)*|**v**|*^{β}^{(}^{x}^{)}*|**u**|*^{α}^{(}^{x}^{)}*u *+ *g*(*x*) *in *Ω*,*

*u *= *v *= 0 *on ∂*Ω*,*

where Ω is an open bounded domains of **R**^{N}* *with a smooth boundary *∂*Ω. The existence of weak solutions is proved using the theory of monotone operators.

* *

*2000 Mathematics Subject Classification. *Primary 35B45; Secondary 35J55.

*Key words and phrases.*Weak solutions; nonlinear elliptic systems;

*p*(

*x*)-Laplacian; monotone operators; generalized Lebesgue-Sobolev spaces.