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

Papageorgiou, Nikolaos S. (author), Rǎdulescu, Vicenţiu (author), Repovš, Dušan (author)

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 ▫$\lambda \mapsto \bar{u}_{\lambda}$▫.

• Language: English
• Keywords: indefinite and unbounded potential, Robin eigenvalue problem, sublinear perturbation, superlinear perturbation, maximum principle, positive solution, minimal positive solution
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2017
• Number of pages: Str. 2589-2618
• Numbering: Vol. 37, no. 5
• PID: 20.500.12556/RUL-109941
• UDC: 517.956
• ISSN on article: 1078-0947
• DOI: http://dx.doi.org/10.3934/dcds.2017111
• COBISS.SI-ID: 17925721
• Publication date in RUL: 10.09.2019

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
• ISSN: 1078-0947
• COBISS.SI-ID: 15865689