We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $��$ varies. We also show that there exists a minimal positive solution $\overline{u}_��$ and determine the monotonicity and continuity properties of the map $��\mapsto\overline{u}_��$. Special attention is given to the particular case of the $p$-Laplacian.

arXiv admin note: text overlap with arXiv:1804.10003