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Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up

Authors: J. García-Melián, R. Letelier-Albornoz and J. Sabina de Lis
Journal: Proc. Amer. Math. Soc. 129 (2001), 3593-3602
MSC (2000): Primary 35J25; Secondary 35B40
Published electronically: June 6, 2001
MathSciNet review: 1860492
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In this paper we prove uniqueness of positive solutions to logistic singular problems $-\Delta u=\lambda(x) u -a(x) u^{p}$, $u_{\vert\partial \Omega }=+\infty $, $p>1$, $a>0$ in $\Omega$, where the main feature is the fact that $a_{\vert\partial\Omega}=0$. More importantly, we provide exact asymptotic estimates describing, in the form of a two-term expansion, the blow-up rate for the solutions near $\partial \Omega $. This expansion involves both the distance function $d(x)=\text{dist}(x,\partial \Omega)$ and the mean curvature $H$ of $\partial \Omega $.

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Additional Information

J. García-Melián
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, Astrofísico Francisco Sánchez s/n, 38271-La Laguna, Spain

R. Letelier-Albornoz
Affiliation: Departamento de Matemáticas, Universidad de Concepción, Casilla 3-C, Concepción, Chile

J. Sabina de Lis
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, Astrofísico Francisco Sánchez s/n, 38271-La Laguna, Spain

Keywords: Boundary blow-up, uniqueness, sub and supersolutions, distance function
Received by editor(s): April 17, 2000
Published electronically: June 6, 2001
Additional Notes: This work was supported by DGES, project PB96-0621 (Spain) and grant FONDECYT No. 1000333 (Chile).
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

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