The nature of critical points generated by a Ghoussoub's type min–max principle for locally Lipschitz continuous functionals fulfilling a weak Palais–Smale assumption, which contains the so-called non-smooth Cerami condition, is investigated. Two meaningful special cases are then pointed out; see Theorems 3.5 and 3.6.

The existence of infinitely many solutions for a Sturm-Liouville boundary value problem, under an appropriate oscillating behavior of the possibly discontinuous nonlinear term, is obtained. Several special cases and consequences are pointed out and some examples are presented. The technical approach is mainly based on a result of infinitely many critical points for locally Lipschitz functions.

In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

The existence of infinitely many solutions for a discrete non-linear Dirichlet problem involving the p-Laplacian, under appropriate oscillating behaviours of the non-linear term, is established. The approach is based on the critical point theory.

: Existence results of two critical points for functionals unbounded from below are established after pointing out a characterization of the mountain pass geometry. Applications to elliptic Dirichlet problems are then presented.