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Semigroups of Linear and Nonlinear Operations and Applications pp 109–119Cite as

Semipositone problems

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Abstract

By a semipositone problem we mean a semilinear equation where the nonlinearity is nondecreasing and negative at the origin. A typical example is the Dirichlet problem

$$\Delta u + \lambda f\left( u \right) = 0{\text{ in }}\Omega ,{\text{ }}u = 0{\text{ on }}\partial \Omega$$
(1.1)

whereλ ∈ (0,∞) is a parameter, Ω is a smooth bounded region in ℝn, Δ is the Laplacian operator and f: ℝ → ℝ is a locally Lipschitzian monotonically increasing function such that

$$ f\left( 0 \right) < 0 $$
(1.2)

and f(σ)> 0 for some σ >0. Semipositone problems naturally arise in various studies.

Supported in part by NSF Grant DMS - 8905936

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Author information

Authors and Affiliations

  1. University of North Texas Denton, TX, 76203, USA

    Alfonso Castro

  2. Mississippi State University, Mississippi State, MS, 39762, USA

    Ratnasingham Shivaji

Authors
  1. Alfonso Castro
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  2. Ratnasingham Shivaji
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana, USA

    Gisèle Ruiz Goldstein  & Jerome A. Goldstein  & 

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© 1993 Springer Science+Business Media Dordrecht

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Castro, A., Shivaji, R. (1993). Semipositone problems. In: Goldstein, G.R., Goldstein, J.A. (eds) Semigroups of Linear and Nonlinear Operations and Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1888-0_4

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  • DOI: https://doi.org/10.1007/978-94-011-1888-0_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4834-7

  • Online ISBN: 978-94-011-1888-0

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