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Resolution of a semilinear equation in L1

Published online by Cambridge University Press:  14 November 2011

Thierry Gallouët  and
Jean-Michel Morel
Thierry Gallouët
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 4 place Jussieu, 75230 Paris Cedex 05, France
Jean-Michel Morel
Affiliation:
Département de Mathématique-Informat, Faculté des Sciences de Luminy, Université d'Aix-Marseille II, 13288 Marseille Cedex 09, France
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Synopsis

Let Ω = ℝN or Ω be a bounded regular open set of ℝN and let γ(x, S): Ω × ℝ → ℝ be a continuous nondecreasing function in s, measurable in x, such that γ(x, 0) = 0 almost everywhere. We solve, for f ∈ L1(Ω), the problem (P): −Δu + γ(., u) = f in Ω, u = 0 on ∂Ω. (In fact, for this result, instead of assuming that γ is nondecreasing in s we need only that γ(x, s)s≧0.) We deduce an ‘almost’ necessary and sufficient condition on , in order that (P) has a solution. Roughly speaking, this condition is f = −ΔV + g, with g ∈ L1(Ω) and γ(., V) ∈ L1(Ω)

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 96 , Issue 3-4 , 1984 , pp. 275 - 288
Copyright
Copyright © Royal Society of Edinburgh 1984

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References

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