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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A strong comparison principle for the -Laplacian


Authors: Paolo Roselli and Berardino Sciunzi
Journal: Proc. Amer. Math. Soc. 135 (2007), 3217-3224
MSC (2000): Primary 35J70; Secondary 35B05
Published electronically: May 14, 2007
MathSciNet review: 2322752
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Abstract: We consider weak solutions of the differential inequality of p-Laplacian type

$\displaystyle - \Delta_p u - f(u) \le - \Delta_p v - f(v)$

such that $ u\leq v$ on a smooth bounded domain in $ \mathbb{R}^N$ and either $ u$ or $ v$ is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that $ u<v$ on the boundary of the domain we prove that $ u<v$, and assuming that $ u\equiv v\equiv0$ on the boundary of the domain we prove $ u < v$ unless $ u \equiv v$. The novelty is that the nonlinearity $ f$ is allowed to change sign. In particular, the result holds for the model nonlinearity $ f(s) = s^q - \lambda s^{p-1} $ with $ q >p-1$.


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Additional Information

Paolo Roselli
Affiliation: Dipartimento di Matematica, Universà di Roma “Tor Vergata”, Via della Ricerca Scientifica 00133 Roma, Italy
Email: roselli@mat.uniroma2.it

Berardino Sciunzi
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
Email: sciunzi@mat.uniroma2.it

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08847-8
PII: S 0002-9939(07)08847-8
Keywords: $p$-Laplace operator, geometric and qualitative properties of the solutions, comparison principle.
Received by editor(s): April 14, 2006
Received by editor(s) in revised form: June 19, 2006
Published electronically: May 14, 2007
Additional Notes: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.