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 $\lambda\mapsto\bar{ u}_{ \lambda}$.
Categories: math.AP