Existence of weak solutions for nonlinear elliptic systems on $R^N$

In this paper, we consider the nonlinear elliptic system $$displaylines{ -Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^alpha|v|^eta v+f,cr -Delta_qv=-c(x)|u|^alpha |v|^eta u + d(x) |v|^{q-2}v +g ,cr lim_{|x|oinfty}u=lim_{|x|oinfty}v=0quad u,v>0 }$$ on a bounded and unbounded domains of mathbb{R}^N, where $Delta_p$ denotes the p-Laplacian. The existence of weak solutions for these systems is proved using the theory of monotone operators