A mountain pass theorem

Abstract

143 In other words, the mountain separating 0 and e may even be assumed of “zero” altitude and the conclusion is still true; moreover (as we shall show) if c = a the critical point can be chosen with llx,,ll = R. It is interesting to ask whether this extension of the Ambrosetti- Rabinowitz theorem remains true in the infinite-dimensional case. We prove here that if (1) is strengthened a little, to the form (1’) there exist real numbers u, r, R such that 0 a for every x E A := {X E X: Y a. Moreover, if c = a the critical point can be chosen with r < I/x0/I < R. Roughly speaking, in brief, the mountain pass theorem continues to hold for a mountain of zero altitude, provided it also has non-zero thickness; in addition, if c = a, the “pass” itself occurs precisely on the mountain-i.e., satisfying r < llroll < R. There are two interesting and immediate corollaries. The first one says that a C’ function which has IWO local minimum points also has a third critical point. The second states that a u-periodic C’ function with a local minimum e has a critical point x,, # e + ku, k = 0, + 1, +2 ,.... The precise statements of the above results will be given in the next sec- tions. The proof depends upon a lemma of Clark [3], formulated here in a version suitable to our purpose. In Section 3 some applications are presen- ted for the forced pendulum equation. 2.