Critical Point Theorems of Non-Smooth Functionals Without the Palais-Smale Condition

Abstract

This paper introduces some new variants of abstract critical point theorems that do not rely on any compactness condition of Palais Smale type. The focus is on locally Lipschitz continuous functional $\Phi$ : $E → R,$ where $E$ is a reflexive banach space. The theorems are established through the utilization of the least action principle, the perturbation argument, the reduction method, and the properties of sub-differential and generalized gradients in the sense of F.H. Clarke. These approaches have been instrumental in advancing the theory of critical points, providing a new perspective that eliminates the need for traditional compactness constraints. The implications of these results are far-reaching, with potential applications in optimization, control theory, and partial differential equations.