Boundary Conditions

In Section 3.4 we saw that there was a distinct advantage to obtaining a bounded Palais-Smale sequence. A method of obtaining such sequences was presented in Section 5.2. In keeping with the mathematical principle “There is no free lunch,” we had to impose a boundary condition on a sphere of radius R in order to produce a bounded Palais-Smale sequence. This boundary condition causes additional restrictions in applications. However, in general, these additional restrictions are more than offset by the requirements that the Palais-Smale conditions be satisfied. In this chapter we shall present some applications in which the boundary condition “pays for itself” in that it imposes no additional restriction and the usual theory does not work without it. In the first application we can apply Theorem 5.2.1 directly. This will be presented in the next section. For the second we shall need a theorem which can be called the direct opposite of Theorem 5.2.1 in that it produces a Palais-Smale sequence completely outside a ball of radius R. This will be presented in Section 9.3. The application will be given in Section 9.4.