Existence of weak solutions for nonlinear elliptic systems involving the (p(x), q(x))-Laplacian

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 $\mathrm{\Omega }$ is an open bounded domains of ${\mathbb{R}}^N$ with a smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ and ${\mathrm{\Delta }}_{p(x)}$ denotes the $p(x)$-Laplacian.The existence of weak solutions is proved using the theory of monotone operators. Similar result will be proved when $\mathrm{\Omega }={\mathbb{R}}^N$.