Some abstract results on the existence of bounded Palais-Smale sequences

Without compactness assumptions, we prove some abstract results which show that a $C^{1}$ functional $I:X\rightarrow \mathbb{R}$ on a Banach space $X$ admits bounded Palais-Smale sequences provided that it exhibits some geometric structure of minimax type and a suitable behaviour with respect to some sequence of continuous mappings $\psi _{n}:X\rightarrow X$. This work is a preliminary version of a forthcoming paper, where applications to nonlinear equations without Ambrosetti-Rabinowitz type assumptions will also be given.