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On the asymptotic properties of solutions to a differential equation in a case of bifurcation without eigenvalues

Published online by Cambridge University Press:  14 November 2011

R. J. Magnus
R. J. Magnus
Affiliation:
Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland
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Synopsis

The semi-linear equation Δu − λu + h(x)uσ = 0 is studied on all of d-dimensional Euclidean space. In the bifurcation problem a non-trivial solution is sought for small λ which tends to zero with λ. The asymptotic dependence of the solution on λ is examined. For fixed λ = 1 the existence of non-degenerate non-trivial solutions is proved for generic measurable h(x) sufficiently near to a constant, provided d = 1 or 3. The two problems are seen to be interdependent. The bifurcation problem at λ = 0 is particularly interesting as the linearised equation is of non-Fredholm type.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 137 - 159
Copyright
Copyright © Royal Society of Edinburgh 1986

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