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The solvability of an elliptic system under a singular boundary condition

Published online by Cambridge University Press:  30 July 2007

J. García-Melián ,
J. Sabina de Lis  and
R. Letelier-Albornoz
J. García-Melián
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, C/Astrofísico Francisco Sánchez s/n, 32871 La Laguna, Spain (jjgarmel@ull.es; josabina@ull.es)
J. Sabina de Lis
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, C/Astrofísico Francisco Sánchez s/n, 32871 La Laguna, Spain (jjgarmel@ull.es; josabina@ull.es)
R. Letelier-Albornoz
Affiliation:
Departamento de Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile
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Abstract

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In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 3 , June 2006 , pp. 509 - 546
Copyright
Copyright © Royal Society of Edinburgh 2006