The Use of Cerami Sequences in Critical Point Theory

Abstract

The concept of linking was developed to produce Palais-Smale (PS) sequences $G\left(u_k\right)\to a$, $G\text{'}\left(u_k\right)\to 0$ for $C^1$functionals $G$ that separate linking sets. These sequences produce critical points if they have convergent subsequences (i.e., if $G$ satisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfying $G\left(u_k\right)\to a$, $(1+|\left|u_k\right|\left|\right)G\text{'}\left(u_k\right)\to 0$ as well. A Cerami sequence can produce a critical point even when a PS sequence does not. In this situation, it is no longer necessary to show that $G$ satisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.