In this paper, we study the existence and nonexistence of solutions for some systems of semilinear elliptic equations in an unbounded domain $\Omega$. The main objective of this paper is to discuss the relationship between the existence of solutions and the translation invariance of the domain $\Omega$. More precisely, we point out how the translation invariance of our domain plays a crucial role for the existence of our problem.
"On the existence and nonexistence of solutions for an elliptic system in unbounded domains." Adv. Differential Equations 12 (7) 759 - 780, 2007.