## 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