Some abstract critical point theorems and applications

Since Palais' pioneer paper in 1963, Condition (C) in both the Palais-Smale version and Cerami's variant has been widely used in order to prove minimax existence theorems for $C^1$ functionals in Banach spaces. Here, we introduce a weaker version of these conditions so that a Deformation Lemma still holds and some critical points theorems can be stated. Such abstract results apply to $p$-Laplacian type elliptic problems.