An extension of the mountain pass lemma

We present a more general form of the mountain pass lemma. It asserts that a C¹ functional which satisfies the Palais–Smale condition admits a critical value when the connectedness of certain level sets changes. We also give an improved form of a theorem given in [A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1) (1981) 1–32], which characterizes the existence of the critical value by means of contractibility properties of the level sets.