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In this book, we study the validity of the maximum principle (one of the most useful and best known tools employed in the study of partial differential equations) for some nonlinear elliptic systems, with variable coefficients, involving the weighted p-Laplcaian operators on bounded and unbounded domains. Also, using a different methods like the nonlinear theory of monotone operators, subsuper solutions method and an approximation (perturbation) method, we study the existence of positive weak solutions for some nonlinear elliptic systems, with variable coefficients, involving the weighted p-Laplcaian operators on bounded and unbounded domains.