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

• Autores: Florica Corina Cîrstea, Yihong Du
• Localización: Proceedings of the London Mathematical Society, ISSN 0024-6115, Vol. 91, Nº 2, 2005, págs. 459-482
• Idioma: inglés
• DOI: 10.1112/s0024611505015273
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• Resumen

  • 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.