  We study perturbations of the eigenvalue problem for the negative Laplacian
plus an indefinite and unbounded potential and Robin boundary condition. First
we consider the case of a sublinear perturbation and then of a superlinear
perturbation. For the first case we show that for
$\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal
eigenvalue) there is one positive solution which is unique under additional
conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$
there are no positive solutions. In the superlinear case, for
$\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for
$\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both
cases we establish the existence of a minimal positive solution
$\bar{u}_{\lambda}$ and we investigate the properties of the map
$\lambda\mapsto\bar{u}_{\lambda}$.
