The Mountain Pass Theorem and Critical Points of Saddle Type

Abstract

In Chapter 9 we shall continue the investigation of the L ^p solutions of the Hammerstein integral equations under the assumption that f (x, 0) = 0, that is, the null function is a solution. We are now interested in non-null solutions. The technique we use is based on the so called mountain pass theorem of Ambrosetti-Rabinowitz [3]. By this method one can establish the existence of a critical point u of the functional E which in general is not an extremum point of E, and has the property that in any neighborhood of u there are points v and w with E (v) < E (u) < E (w). Such a critical point is said to be a saddle point of E.