Quantitative deformation theorems and critical point theory

It is well known that deformation theorems are the basic tools in critical point theory. They can be derived under a condition of Palais-Smale type ((PS), for short). In the classical setting of a C1 functional f defined on a Banach space X (or a C2 Finsler manifold), we refer to [15]; for a continuous functional f defined on a complete metric space X, we refer to [8], the results of which include the case of a C1 functional defined on a C1 Finsler manifold. On the other hand, some authors gave, in the smooth case, so-called quantitative deformation theorems which are based on a weaker condition than (PS), see, e.g., [2, 21], and also [16] for a detailed account of the theory. Now, it is also well known how the min-max principle yields the existence of a critical value of the functional f, by combining a deformation theorem under the (PS) condition, and some geometrical assumptions. Generally speaking, this is the case when a subset B of X links another subset A of X, and the condition −∞ < b0 := sup B f ≤ inf A f =: a < +∞ (1.1)