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

Abstract

Without compactness assumptions, we prove some abstract results which show that a $C^1$ functional $I:X\to \mathbb{R}$ 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 ${\psi }_n:X\to X$. This work is a preliminary version of a forthcoming paper, where applications to nonlinear equations without Ambrosetti-Rabinowitz type assumptions will also be given.

arXiv commentThis document is a preliminary version of a forthcoming paper, where applications to nonlinear equations will be added