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Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity

Published online by Cambridge University Press:  12 July 2007

José M. Arrieta
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain (arrieta@mat.ucm.es; rpardo@mat.ucm.es; arober@mat.ucm.es)
Rosa Pardo
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain (arrieta@mat.ucm.es; rpardo@mat.ucm.es; arober@mat.ucm.es)
Anibal Rodríguez-Bernal
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain (arrieta@mat.ucm.es; rpardo@mat.ucm.es; arober@mat.ucm.es)

Abstract

We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2007

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