In this article we consider the system of equations Δu_i = p_i(x)f_i(u₁,...,u_d) for i = 1,...,d on R^N, N ≥ 3 and d ∈ {1,2,3,4,...}. We prove that the considered system has a bounded positive entire solution under some conditions on p_i and f_i. Also, we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation. Furthermore, a non-radially symmetric solution is obtained by using a lower and upper solution method.