bounded palais–smale sequences for non-differentiable functions

the existence of bounded palais–smale sequences (briefly bps) for functionals depending on a parameter belonging to a real interval and which are the sum of a locally lipschitz continuous term and of a convex, proper, lower semicontinuous function, is obtained when the parameter runs in a full measure subset of the given interval. specifically, for this class of non-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.