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Purpose – In this paper, the authors give a new version of the sub-super solution method and prove the existence of positive solution for a (p, q)-Laplacian system under weak assumptions than usually made in such systems. In particular, nonlinearities need not be monotone or positive. Design/methodology/approach – The authors prove that the sub-super solution method can be proved by the Shcauder fixed-point theorem and use the method to prove the existence of a positive solution in elliptic systems, which appear in some problems of population dynamics. Findings – The results complement and generalize some results already published for similar problems. Originality/value – The result is completely new and does not appear elsewhere and will be a reference for this line of research.