Geometric description of the mountain pass critical points

In this short commentary I present the main results contained in the famous paper [P. Pucci & J. Serrin, The structure of the critical set in the mountain pass theorem, Trans. Amer. Math. Soc. 299 (1987), no. 1, 115–132]. This paper deals with critical points obtained via the mountain pass lemma of A. Ambrosetti and P.H. Rabinowitz. The structure of such critical points is studied with a remarkable precision and several general results are obtained, such as, for example, in the case of an infinite-dimensional problem, a general alternative between saddle points of mountain pass type or a set of local minima whose closure intersects at least two components of the set of saddle points. The questions which arose in the paper inspired a number of further research papers and the commentary lists the applications.