LAUSR

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 \begin{document}$ λ ( \begin{document}$ \widehat{λ}_{1}$\end{document} being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For \begin{document}$ λ≥q\widehat{λ}_{1}$\end{document} there are no positive solutions. In the superlinear case, for \begin{document}$ λ we have at least two positive solutions and for \begin{document}$ λ≥q\widehat{λ}_{1}$\end{document} there are no positive solutions. For both cases we establish the existence of a minimal positive solution \begin{document}$ \bar{u}_{λ}$\end{document} and we investigate the properties of the map \begin{document}$ λ\mapsto\bar{u}_{λ}$\end{document} .