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

TL;DR

This paper establishes abstract conditions under which bounded Palais-Smale sequences exist for certain functionals on Banach spaces, without relying on compactness assumptions, and discusses potential applications to nonlinear equations.

Contribution

It introduces new abstract criteria for the existence of bounded Palais-Smale sequences without compactness, extending previous results and setting the stage for applications to nonlinear equations.

Findings

Proves existence of bounded Palais-Smale sequences under geometric and behavioral conditions.

Develops a framework applicable to nonlinear equations without Ambrosetti-Rabinowitz conditions.

Lays groundwork for future applications to nonlinear analysis.

Abstract

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.