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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type
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by Florica-Corina Cîrstea and Vicenţiu Rădulescu PDF
Trans. Amer. Math. Soc. 359 (2007), 3275-3286 Request permission

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
  • Email: Florica.Cirstea@maths.anu.edu.au
  • Vicenţiu Rădulescu
  • Affiliation: Department of Mathematics, University of Craiova, 200585 Craiova, Romania
  • MR Author ID: 143765
  • ORCID: 0000-0003-4615-5537
  • Email: radulescu@inf.ucv.ro
  • 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.
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2299455