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A uniqueness result for a singular elliptic equation with gradient term

  • José Carmona (a1) and Tommaso Leonori (a2)

Abstract

We prove the uniqueness of a solution for a problem whose simplest model is

with k ≥ 1, 0 f ∈ L(Ω) and Ω is a bounded domain of ℝN, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

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Present address: Dipartimento di Scienze di Base e Applicate per l'Ingegneria ‘Sapienza’, Università di Roma I, Via Antonio Scarpa 10, 00161 Roma, Italy (tommaso.leonori@sbai.uniroma1.it).

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* Present address: Dipartimento di Scienze di Base e Applicate per l'Ingegneria ‘Sapienza’, Università di Roma I, Via Antonio Scarpa 10, 00161 Roma, Italy ().

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