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Existence of positive solutions for a class of semipositone problem in whole ℝN

Part of: Partial differential equations Qualitative behavior Elliptic equations and systems

Published online by Cambridge University Press:  05 April 2019

Claudianor O. Alves ,
Angelo R. F. de Holanda  and
Jefferson A. dos Santos
Claudianor O. Alves
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande - PB, Brazil (coalves@mat.ufcg.edu.br); (angelo@mat.ufcg.edu.br)(jefferson@mat.ufcg.edu.br)
Angelo R. F. de Holanda
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande - PB, Brazil (coalves@mat.ufcg.edu.br); (angelo@mat.ufcg.edu.br)(jefferson@mat.ufcg.edu.br)
Jefferson A. dos Santos
Affiliation:
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática, CEP: 58429-900, Campina Grande - PB, Brazil (coalves@mat.ufcg.edu.br); (angelo@mat.ufcg.edu.br)(jefferson@mat.ufcg.edu.br)
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Abstract

In this paper we show the existence of solution for the following class of semipositone problem P

$$\left\{\matrix{-\Delta u & = & h(x)(f(u)-a) & \hbox{in} & {\open R}^N, \cr u & \gt & 0 & \hbox{in} & {\open R}^N, \cr}\right.$$
where N ≥ 3, a > 0, h : ℝN → (0, + ∞) and f : [0, + ∞) → [0, + ∞) are continuous functions with f having a subcritical growth. The main tool used is the variational method together with estimates that involve the Riesz potential.

Keywords

Variational methodssemipositone problempositive solutionsRiesz potential

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 45M20: Positive solutions 35A08: Fundamental solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2349 - 2367
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Copyright
Copyright © Royal Society of Edinburgh 2019

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