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Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

  • * Corresponding author: Vicenţiu D. Rădulescu

    * Corresponding author: Vicenţiu D. Rădulescu 
This research was supported in part by the Slovenian Research Agency grants P1-0292, J1-7025 and J1-6721. V.D. Rădulescu was also supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-Ⅱ-PT-PCCA-2013-4-0614.
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  • 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 $ λ<\widehat{λ}_{1}$ ($ \widehat{λ}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. In the superlinear case, for $ λ<\widehat{λ}_{1}$ we have at least two positive solutions and for $ λ≥q\widehat{λ}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $ \bar{u}_{λ}$ and we investigate the properties of the map $ λ\mapsto\bar{u}_{λ}$.

    Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60.

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