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We prove that for any λ ∈ IR, there is a sequence of eigencurves (µk(λ))k for the nonlinear coupled elliptic system
by using min-max methods. We prove also via an homogeneity type condition that the eigenvector corresponding to the principal µ1(λ) is bounded, positive and smooth. We end this work by proving that µ1(λ) is simple.