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For N≥ 3 and non-negative real numbers a_ij and b_ij (i,j= 1, …, m), the semi-linear elliptic systemequation* casesΔ u_i+_j=1^m u_j^a_ij =0,inR_+^N, u_iy_N=c_i_j=1^m u_j^b_ij ,on R_+^N,casesi=1,…,m,equation* % is considered, where R_+^N is the upper half of N-dimensional Euclidean space. Under suitable assumptions on the exponents a_ij and b_ij, a classification theorem for the positive C^2(R_+^N)C^1(R_+^N)-solutions of this system is proven. , number=1, journal=Journal of Partial Differential Equations, year=2015, month=Mar., pages=74–94