The paper deals with the existence of nonnegative solutions for (p,N) systems in R^N involving critical exponential growth nonlinearities. The constructed solution has both components nontrivial and different, that is it solves the actual system, which does not reduce into an equation. The main feature and novelty of the paper is the presence of a general coupled critical exponential term of the Trudinger-Moser type, set in R^N
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