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A uniqueness result for a singular elliptic equation with gradient term

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  22 June 2018

José Carmona  and
Tommaso Leonori
José Carmona
Affiliation:
Departamento de Matemáticas, Universidad de Almería, Carretera Sacramento S/N, La Cañada de San Urbano, 04120 Almería, Spain (jcarmona@ual.es)
Tommaso Leonori
Affiliation:
Departamento de Análisis Matemático, Campus Fuentenueva S/N, Universidad de Granada, 18071 Granada, Spain (leonori@ugr.es)
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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.

Keywords

nonlinear elliptic equationssingular natural growth gradient termscomparison principle

MSC classification

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations 35J75: Singular elliptic equations
Secondary: 35A02: Uniqueness problems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 5 , October 2018 , pp. 983 - 994
Copyright
Copyright © Royal Society of Edinburgh 2018 

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Footnotes

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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).