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.
– 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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