Let \(X\) be an infinite dimensional Banach space. This paper is devoted to the study of critical points for functionals of the form \(f=\varphi +\alpha\), where \(\varphi :X\rightarrow \mathbb R\) is locally Lipschitz continuous and \(\alpha :X\rightarrow \mathbb R\cup\{+\infty\}\) is convex, proper and lower semicontinuous. Under some natural assumptions, the main abstract result of the paper establishes the existence of a critical point of \(f\). The corresponding theorem extends a previous result obtained in [S. Adly, G. Buttazzo and M. Théra, “Critical points for nonsmooth energy functions and applications”, Nonlinear Anal., Theory Methods Appl. 32, No. 6, 711–718 (1998; Zbl 0940.49019)]. This result is then applied for solving a hemivariational inequality involving a subcritical nonlinearity.
The proofs employ refined arguments related to the critical point theory in the sense of Clarke and Chang. The reviewer remarks that the techniques developed in this paper can be extended for solving other classes of nonlinear inequality problems.