A Deformation Theorem and Some Critical Point Results for Non-differentiable Functions

A deformation lemma for functionals which are the sum of a locally Lipschitz continuous function and of a concave, proper and upper semicontinuous function is established. Some critical point theorems are then deduced and an application to a class of elliptic variational-hemivariational inequalities is presented. Introduction It is by now well known that the Mountain Pass Theorem of Ambrosetti and Rabinowitz [2, Theorem 2.1] employs fruitfully in the study of various questions concerning differential equations. This result basically applies to each case when the solutions of the problem under consideration can be regarded as critical points of a continuously differentiable real-valued functional f on a Banach space (X, ‖ · ‖), with the following property: (f) there exist x0, x1 ∈ X, r > 0, a ∈ R such that ‖x1 − x0‖ > r and max{f(x0), f(x1)} < a ≤ f(x) for all x ∈ ∂B(x0, r), where ∂B(x0, r) = {x ∈ X : ‖x− x0‖ = r}. 2000 Mathematics Subject Classification. Primary 35A15, 49J40, 35J85.