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Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN

Published online by Cambridge University Press:  14 November 2011

Hans-Peter Heinz
Hans-Peter Heinz
Affiliation:
Fachbereich Mathematik der JohannesGutenberg-Universitat Mainz, Postfach 3980, D-6500 Mainz, Germany
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Synopsis

We consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general case and bifurcation results for nonlinear perturbations of the periodic Schrödinger equation.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 118 , Issue 3-4 , 1991 , pp. 237 - 270
Copyright
Copyright © Royal Society of Edinburgh 1991

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