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Boundary blow-up in nonlinear elliptic equations of Bieberbach-Rademacher type


Authors: Florica-Corina Cîrstea and Vicentiu Radulescu
Journal: Trans. Amer. Math. Soc. 359 (2007), 3275-3286
MSC (2000): Primary 35J25; Secondary 35B40, 35J60
DOI: https://doi.org/10.1090/S0002-9947-07-04107-4
Published electronically: February 13, 2007
MathSciNet review: 2299455
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Abstract: We establish the uniqueness of the positive solution for equations of the form $ -\Delta u=au-b(x)f(u)$ in $ \Omega$, $ u\vert _{\partial\Omega}=\infty$. The special feature is to consider nonlinearities $ f$ whose variation at infinity is not regular (e.g., $ \exp(u)-1$, $ \sinh(u)$, $ \cosh(u)-1$, $ \exp(u)\log(u+1)$, $ u^\beta \exp(u^\gamma)$, $ \beta\in {\mathbb{R}}$, $ \gamma>0$ or $ \exp(\exp(u))-e$) and functions $ b\geq 0$ in $ \Omega$ vanishing on $ \partial\Omega$. The main innovation consists of using Karamata's theory not only in the statement/proof of the main result but also to link the nonregular variation of $ f$ at infinity with the blow-up rate of the solution near $ \partial\Omega$.


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Additional Information

Florica-Corina Cîrstea
Affiliation: Department of Mathematics, The Australian National University, Canberra, ACT 0200, Australia
Email: Florica.Cirstea@maths.anu.edu.au

Vicentiu Radulescu
Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
Email: radulescu@inf.ucv.ro

DOI: https://doi.org/10.1090/S0002-9947-07-04107-4
Keywords: Large solutions, boundary blow-up, regular variation theory
Received by editor(s): April 16, 2004
Received by editor(s) in revised form: May 11, 2005
Published electronically: February 13, 2007
Additional Notes: The research of the first author was carried out at Victoria University (Melbourne) with the support of the Australian Government through DETYA
The second author has been supported by Grant 2-CEX06-11-18/2006.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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