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.

Bifurcation from the essential spectrum

Existence and Bifurcation Theorems for Nonlinear Elliptic Eigenvalue Problems on Unbounded Domains

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Bifurcation from the Essential Spectrum

Nodal properties and bifurcation from the essential spectrum for a class of nonlinear Sturm-Liouville problems

A class of nonlinear Sturm-Liouville problems is considered. These problems admit zero as a trivial solution and the nonlinear operator linearized about zero has a purely continuous spectrum [0, ∞). Uniqueness of solutions with a prescribed number of nodes is obtained by a shooting argument. Moreover, combining with the previous existence results, we prove there are infinitely many continua of solutions, which are distinguished by nodal properties, bifurcate from the line of trivial solutions at the lowest point of the continuous spectrum.

Uniqueness and bifurcation for solutions of nonlinear Sturm-Liouville eigenvalue problems

Bifurcation from the continuous spectrum of a linearized operator is of interest in many physical problems. For example it occurs in the nonlinear Klein-Gordon equation and in nonlinear integrodifferential equations as the Choquard problem; it further appears in nonlinear integral equations of the convolution type.



A general theory enclosing all these problems is not yet known. To understand the basic phenomena, we therefore consider monotone differential operators whose linearisations have a purely continuous spectrum. It is shown that in fact the lowest point of the continuous spectrum is a bifurcation point, if the nonlinearity grows sufficiently strong.

G. F. Roach

Papers

On minimal nonlinearities which permit bifurcation from the continuous spectrum