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Proceedings of the London Mathematical Society Advance Access published online on November 27, 2006
Proceedings of the London Mathematical Society, 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
2 Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina, josabina{at}ull.es
3 Instituto de Matemáticas y Fí sica 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 {Omega} 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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