Bounded Palais-Smale sequence for non differentiable functions

The existence of bounded Palais–Smale sequences (briefly BPS) for functionals dependingon a parameter belonging to a real interval and which are the sum of a locally Lipschitzcontinuous term and of a convex, proper, lower semicontinuous function, is obtained whenthe parameter runs in a full measure subset of the given interval. Specifically, for this class ofnon-smooth functions, we obtain BPS related to mountain pass and to global infima levels.This is done by developing a unifying approach, which applies to both cases and relies on a suitable deformation lemma.