A specific feature of many nonlinear problems is the existence of multiple solutions and often it is useful to introduce a parameter ? to detect when new solutions arise. From the mathematical point of view, one is led to consider a functional equation S( ?, u) = 0, depending on a parameter ?, and such that S( ?, 0) ? 0. Bifurcation theory deals with the existence of values ?* at which nontrivial solutions branch off from the trivial one, u = 0. A very interesting survey on bifurcation theory is contained in the paper [145] by G. Prodi, which also contains applications to elasticity and fluid dynamics.

In this chapter we will address the simplest situation, the bifurcation from a simple eigenvalue. The material discussed in this chapter is closely related to that contained in [20], Chapter 5.

2.1 Bifurcation: Definition and Necessary Conditions

Let X, Y be Banach spaces. We will deal with an equation like

where S : X ? Y is such that

The solution u = 0 will be called the trivial solution of (2.1). The set

will be called the set of nontrivial solutions of (2.1). When no confusion is possible, we will omit the subscript S.

Many problems arising in applications...