Bifurcation from the Essential Spectrum

This report surveys some of the results that have been obtained during the past twenty years concerning bifurcation from a point in the essential spectrum of the linearization of a nonlinear equation. This means that the linearization is not a Fredholm operator and so, even locally, the problem cannot be reduced to an equivalent finite dimensional situation by the Lyapunov-Schmidt method. Nonetheless, the aim is to obtain conclusions, local or global, about bifurcation in the same spirit as the classical results which are well-known in the Fredholm case. A vast survey of recent progress in bifurcation theory for the Fredholm situation has been given by Ize [30] in an article in the previous volume of this series. In the present context all of the standard techniques of nonlinear analysis (variational methods, topological degree, implicit function theorems) have been brought to bear on the problem, but so far only the variational approach has yielded general results in abstract spaces. The other methods have been confined to the context of elliptic equations on unbounded domains or to integral equations involving convolution. This state of affairs is reflected in the presentation of this survey which concentrates on the general results obtained by variational methods and their application to elliptic equations on ℜ N . However in the last section there are a few remarks covering what is known about connected sets, or even curves, of solutions to differential equations and it is to be hoped that in the near future substantial progress will be made in establishing conclusions of this kind in an abstract setting similar to that used in the variational case.

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