Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential
Papageorgiou, Nikolaos (Author), Rǎdulescu, Vicenţiu (Author), Repovš, Dušan (Author)

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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 ▫$\lambda < \widehat{\lambda}_{1}$▫ (▫$\widehat{\lambda}_{1}$▫ 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 ▫$\lambda \mapsto \bar{u}_{\lambda}$▫.

Keywords:indefinite and unbounded potential, Robin eigenvalue problem, sublinear perturbation, superlinear perturbation, maximum principle, positive solution, minimal positive solution
Work type:Article (dk_c)
Tipology:1.01 - Original Scientific Article
Organization:PEF - Faculty of Education
FMF - Faculty of Mathematics and Physics
Number of pages:Str. 2589-2618
Numbering:Vol. 37, no. 5
ISSN on article:1078-0947
DOI: http://dx.doi.org/10.3934/dcds.2017111 This link opens in a new window
COBISS.SI-ID:17925721 This link opens in a new window
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Record is a part of a journal

Title:Discrete and continuous dynamical systems
Shortened title:Discrete contin. dyn. syst.
Publisher:American Institute of Mathematical Sciences
COBISS.SI-ID:15865689 This link opens in a new window

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