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Bifurcation from infinity for reaction–diffusion equations under nonlinear boundary conditions

Published online by Cambridge University Press:  20 March 2017

Nsoki Mavinga
Affiliation:
Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA 19081-1390, USA ([email protected])
Rosa Pardo
Affiliation:
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain ([email protected])

Extract

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2017 

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