Critical Point Theory

In this chapter we present the main variational tool in the study of boundary value problems, which is the critical point theory. In Sect. 3.1 we introduce the compactness conditions on the functional which compensate for the fact that the ambient space is not locally compact (being infinite dimensional) and also a replacement of the gradient when the gradient flow can not be used. With these tools in Sect. 3.2 we prove the deformation theorems which in Sect. 3.3 produce the minimax theorems for the critical values of \(\varphi \) .