Positive solutions for a class of singular Dirichlet problems

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

We consider a Dirichlet elliptic problem driven by the Laplacian with singular and superlinear nonlinearities. The singular term appears on the left-hand side while the superlinear perturbation is parametric with parameter ▫$\lambda > 0$▫ and it need not satisfy the AR-condition. Having as our starting point the work of Diaz-Morel-Oswald (1987) [J.I. Diaz, J.M. Morel, L. Oswald, An elliptic equation with singular nonlinearity, Commun. Partial Differ. Equ. 12 (1987) 1333-1344], we show that there is a critical parameter value ▫$\lambda_\ast$▫ such that for all ▫$\lambda > \lambda_\ast$▫ the problem has two positive solutions, while for ▫$\lambda < \lambda_\ast$▫ there are no positive solutions. What happens in the critical case ▫$\lambda = \lambda_\ast$▫ is an interesting open problem.

• Language: English
• Keywords: singular term, superlinear perturbation, weak comparison, order cone
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2019
• Number of pages: Str. 6539-6554
• Numbering: Vol. 267, iss. 11
• PID: 20.500.12556/RUL-110524
• UDC: 517.956.2
• ISSN on article: 0022-0396
• DOI: 10.1016/j.jde.2019.07.018
• COBISS.SI-ID: 18687833
• Publication date in RUL: 16.09.2019

Record is a part of a journal

• Title: Journal of differential equations
• Shortened title: J. differ. equ.
• Publisher: Elsevier
• ISSN: 0022-0396
• COBISS.SI-ID: 25730560