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Proceedings of the London Mathematical Society Advance Access originally published online on November 27, 2006
Proceedings of the London Mathematical Society 2007 94(1):1-25; doi:10.1112/plms/pdl001
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© 2006 London Mathematical Society

A bifurcation problem governed by the boundary condition II

Jorge García-Melián1, Julio D. Rossi2,3 and José C. Sabina De Lis1

1 Departamento de Análisis Matemático
Universidad de La Laguna
C/ Astrofísico Francisco Sánchez s/n
38271 La Laguna
Spain
jjgarmel{at}ull.es
josabina{at}ull.es
2 Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
1428 Buenos Aires
Argentina
3 Instituto de Matemáticas y Física Fundamental
CSIC
C/ Serrano 123
28006 Madrid
Spain
jrossi{at}dm.uba.ar

Received 4 May 2005. Revision received 17 January 2006.

In this work we consider the problem {Delta}u = a(x)up in Formula on {partial}{Omega}, where {Omega} is a smooth bounded domain, {nu} is the outward unit normal to {partial}{Omega}, {lambda} is regarded as a parameter and 0 < p < 1. We consider both cases where a(x) > 0 in {Omega} or a(x) is allowed to vanish in a whole subdomain {Omega}0 of {Omega}. Our main results include existence of non-negative non-trivial solutions in the range 0 < {lambda} < {sigma}1, where {sigma}1 is characterized by means of an eigenvalue problem, uniqueness and bifurcation from infinity of such solutions for small {lambda}, and the appearance of dead cores for large enough {lambda}.


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