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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Florica-Corina Cîrstea and Vicenţiu Rădulescu
Journal: Trans. Amer. Math. Soc. 359 (2007), 3275-3286
MSC (2000): Primary 35J25; Secondary 35B40, 35J60
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 $\Omega$, $u|_{\partial \Omega }=\infty$. 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

Vicenţiu Rădulescu
Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
MR Author ID: 143765
ORCID: 0000-0003-4615-5537

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.