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On an elliptic problem with boundary blow-up and a singular weight: the radial case

Published online by Cambridge University Press:  12 July 2007

M. Chuaqui ,
C. Cortázar ,
M. Elgueta ,
C. Flores ,
R. Letelier  and
J. García-Melián
M. Chuaqui
Affiliation:
Facultad de Matemáticas, Universidad Católica de Chile, Casilla 306, correo 22—Santiago, Chilemchuaqui@mat.puc.cl
C. Cortázar
Affiliation:
Facultad de Matemáticas, Universidad Católica de Chile, Casilla 306, correo 22—Santiago, Chileccortaza@mat.puc.cl
M. Elgueta
Affiliation:
Facultad de Matemáticas, Universidad Católica de Chile, Casilla 306, correo 22—Santiago, Chilemelgueta@mat.puc.cl
C. Flores
Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 3-C—Concepción, Chilecflores@gauss.cfm.udec.cl
R. Letelier
Affiliation:
Departamento de Matemática, Universidad de Concepción, Casilla 3-C—Concepción, Chilejjgarmel@ull.es
J. García-Melián
Affiliation:
Departamento de Análisis Matemáico, Universidad de La Laguna, c/. Astrofísico Francisco Sánchez s/n, 38271—La Laguna, Spain; and Centro de Modelamiento Matemático, Universidad de Chile, Blanco Encalada 2120, 7 piso—Santiago, Chile (rletelie@gauss.cfm.udec.cl)
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Abstract

In this work we consider the non-autonomous problem Δu = a(x)um in the unit ball B ⊂ RN, with the boundary condition u|∂B = +∞, and m > 0. Assuming that a is a continuous radial function with a(x) ˜ C0 dist(x, ∂B)−γ as dist(x, ∂B) → 0, for some C0 > 0, γ > 0, we completely determine the issues of existence, multiplicity and behaviour near the boundary for radial positive solutions, in terms of the values of m and γ. The case 0 < m ≤ 1, as well as estimates for solutions to the linear problem m = 1, are a significant part of our results.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 133 , Issue 6 , December 2003 , pp. 1283 - 1297
Copyright
Copyright © Royal Society of Edinburgh 2003

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