Nikolaos Socrates Papageorgiou (Author), Vicenţiu Rǎdulescu (Author), Dušan Repovš (Author)

Abstract

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;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
UDC: 517.956
COBISS: 17925721 Link will open in a new window
ISSN: 1078-0947
Views: 456
Downloads: 273
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Other data

Type (COBISS): Article
Pages: str. 2589-2618
Volume: ǂVol. ǂ37
Issue: ǂno. ǂ5
Chronology: 2017
DOI: http://dx.doi.org/10.3934/dcds.2017111
ID: 11221111