The structure of the critical set in the general mountain pass principle (Q1894799)

Let \(X\) be a Banach space and \(\varphi\in C^1(X, R)\) a functional satisfying the Palais-Smale condition. If \(u\), \(v\) are two distinct points in \(X\) and there exists a closed set \(F\) which separates them and is such that \(\varphi(x)\geq \max\{\varphi(u), \varphi(v)\}\) for all \(x\in F\), then \(\varphi\) has a critical point different from \(u\), \(v\). This is (an extension of) the well-known mountain pass theorem of Ambrosetti and Rabinowitz. The paper studies the structure of the set \(K_c\) of critical points on the mountain pass level. Several results of Hofer, Pucci-Serrin and Ghoussoub-Preiss are extended. The main result states that if \(u\), \(v\) are not critical points then \(K_c\) contains either a set of proper local maxima which separates \(u\) and \(v\), or a saddle point of mountain pass type, or at least two disjoint sets of saddle points which are connected through a set (in \(K_c\)) of local minima.