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General uniqueness results and variation speed for blow-up solutions of elliptic equations

Published online by Cambridge University Press:  23 August 2005

Florica Corina Cîrstea  and
Yihong Du
Florica Corina Cîrstea
Affiliation:
School of Computer Science and Mathematics, Victoria University of Technology, PO Box 14428, Melbourne, VIC 8001, Australia Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia. E-mail: Florica.Cirstea@maths.anu.edu.au
Yihong Du
Affiliation:
School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia and Department of Mathematics, Qufu Normal University, P.R. China. E-mail: ydu@turing.une.edu.au, http://mcs.une.edu.au/~ydu/
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Abstract

Let $\Omega$ be a smooth bounded domain in ${R}^N$. We prove general uniqueness results for equations of the form $- \Delta u = au - b(x) f(u)$ in $\Omega$, subject to $u = \infty$ on $\partial \Omega$. Our uniqueness theorem is established in a setting involving Karamata's theory on regularly varying functions, which is used to relate the blow-up behavior of $u(x)$ with $f(u)$ and $b(x)$, where $b \equiv 0$ on $\partial \Omega$ and a certain ratio involving $b$ is bounded near $\partial \Omega$. A key step in our proof of uniqueness uses a modification of an iteration technique due to Safonov.

Keywords

boundary blow-upelliptic equationregularly varying functions35J25 (primary)35B4035J60 (secondary)
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 91 , Issue 2 , September 2005 , pp. 459 - 482
Copyright
2005 London Mathematical Society

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Footnotes

F. Cîrstea was supported by the Australian Government through DETYA under the IPRS Programme. Y. Du was partially supported by the Australian Research Council.