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Article Dans Une Revue Bollettino dell'Unione Matematica Italiana Année : 2009

An elliptic problem with a lower order term having singular behaviour

Résumé

We prove the existence of distributional solutions to an elliptic problem with a lower order term which depends on the solution u in a singular way and on its gradient Du with quadratic growth. The prototype of the problem under consideration is
– Δu + λu = ± |Du |2 / |u|k + ƒ in Ω, u = 0 on ∂Ω,
where  λ  >  0, k  >  0, ƒ  ∈  L(Ω),  ƒ(x)  ≥  0 (and so u ≥ 0). If 0  < k  < 1, we prove the existence of a solution for both the “+” and the “-” signs, while if k ≥ 1, we prove the existence of a solution for the “+” sign only.
This paper has been published in Boll. Un. Mat. Ital., 2, (2009), pp. 349-370.
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Dates et versions

hal-00399023 , version 1 (25-06-2009)

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  • HAL Id : hal-00399023 , version 1

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Daniela Giachetti, François Murat. An elliptic problem with a lower order term having singular behaviour. Bollettino dell'Unione Matematica Italiana, 2009, 2 (2009). ⟨hal-00399023⟩
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