Variational Method

Abstract

In this chapter, we consider problems which are derived from a functional on a function space in variational problems. We will begin with an intro­duction of existence theory by using minimization techniques and some elementary facts about monotone operators. By using a compactness con­dition (condition (C)), we give a minimax principle in Banach spaces. This result will then be used to prove a theorem (mountain pass theorem) on the existence of critical points which are not necessary minimal points. An application to a semi-linear hyperbolic equation will be considered. In Sections 4.8 and 4.9, we present a summary of Ljusternik-Schnirelman theory on Banach manifolds with an application.