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The sublinear problem for the $ 1$-homogeneous $ p$-Laplacian


Authors: Pedro J. Martínez-Aparicio, Mayte Pérez-Llanos and Julio D. Rossi
Journal: Proc. Amer. Math. Soc. 142 (2014), 2641-2648
MSC (2010): Primary 35A02, 35B51, 35J60
DOI: https://doi.org/10.1090/S0002-9939-2014-12108-3
Published electronically: April 21, 2014
MathSciNet review: 3209320
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Abstract: In this paper we prove the existence and uniqueness of a positive viscosity solution of the $ 1$-homogeneous $ p$-Laplacian with a sublinear right-hand side; that is, $ -\vert D u\vert^{2-p}{\rm div}\:(\vert D u\vert^{p-2}Du)=\lambda u^q$ in $ \Omega $, $ u=0$ on $ \partial \Omega $, where $ \Omega $ is a bounded starshaped domain, $ \lambda >0$, $ p>2$ and $ 0<q<1$.


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Pedro J. Martínez-Aparicio
Affiliation: Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, 30202, Murcia, Spain
Email: pedroj.martinez@upct.es

Mayte Pérez-Llanos
Affiliation: Departamento de Matemáticas, Campus de Cantoblanco, Universidad Autonoma de Madrid, 28049, Madrid, Spain
Email: mayte.perez@uam.es

Julio D. Rossi
Affiliation: Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, 03080 Alicante, Spain (on leave from Departamento de Matemática, FCEyN UBA, Ciudad Universitaria, Pab 1 (1428), Buenos Aires, Argentina)
Email: julio.rossi@ua.es

DOI: https://doi.org/10.1090/S0002-9939-2014-12108-3
Keywords: Infinity-Laplacian, $1$-homogeneous $p$-Laplacian
Received by editor(s): September 23, 2011
Published electronically: April 21, 2014
Additional Notes: The first author was supported by MICINN Ministerio de Ciencia e Innovacion (Spain) MTM2009-10878 and Junta de Andalucia FQM-116
The second and third authors were supported by project MTM2010-18128 (Spain)
Communicated by: James E. Colliander
Article copyright: © Copyright 2014 American Mathematical Society