Geodesic
Existence of weak solutions for nonlinear elliptic systems on \(\mathbb{R}^N\)
Electronic journal of differential equations, Tome 2006 (2006)
In this paper, we consider the nonlinear elliptic system

$\displaylines{ -\Delta_pu=a(x)|u|^{p-2}u-b(x)|u|^\alpha|v|^\beta v+f,\cr -\Delta_qv=-c(x)|u|^\alpha |v|^\beta u + d(x) |v|^{q-2}v +g ,\cr \lim_{|x|\to\infty}u=\lim_{|x|\to\infty}v=0\quad u,v>0 }$

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
Classification : 35B45, 35J55
Keywords: weak solutions, nonlinear elliptic systems, p-Laplacian, monotone operators 
@article{EJDE_2006__2006__a45,
     author = {El-Zahrani,  Eada A. and Serag,  Hassan M.},
     title = {Existence of weak solutions for nonlinear elliptic systems on {\(\mathbb{R}^N\)}},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1128.35328},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a45/}
}
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El-Zahrani,  Eada A.; Serag,  Hassan M. Existence of weak solutions for nonlinear elliptic systems on \(\mathbb{R}^N\). Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a45/