  In this paper, we prove the existence of weak solutions for the following
nonlinear elliptic system {lll} -\Delta_{p(x)}u = a(x)|u|^{p(x)-2}u -
b(x)|u|^{\alpha(x)}|v|^{\beta(x)} v + f(x) in \Omega, \Delta_{q(x)}v = c(x)
|v|^{q(x)-2}v - d(x)|v|^{\beta(x)}|u|^{\alpha(x)} u + g(x) in \Omega, u = v = 0
\quad on \partial\Omega, where $\Omega$ is an open bounded domains of
$\mathbb{R}^N$ with a smooth boundary $\partial\Omega$ and $\Delta_{p(x)}$
denotes the $p(x)$-Laplacian.The existence of weak solutions is proved using
the theory of monotone operators. Similar result will be proved when
$\Omega=\mathbb{R}^N$.
