In this work, we study the existence of the principal eigencurves for a Steklov problem with an indefinite weight for homogeneous perturbation of the p-Laplacian operator. We then establish many properties of these eigencurves: continuity, differentiability and asymptotic behaviors. We also use our approach to get similar result when mixed Dirichlet-Steklov boundary condition is considered.

Abstract. In this work, we discuss the existence and the non-existence of principal eigenvalue in an unbounded domain of R N for some potentials which change sign. We also give certain properties of this principal eigenvalue.

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