On the Palais-Smale condition for action integrals (Q1185740)

The author studies the Palais-Smale condition for the action integral \[ f(u) = \int^ 1_ 0 \{\frac 12 | \dot{u}|^ 2 - V(u)\} dt, \quad u\in H^ 1(\mathbb{R}/\mathbb{Z},\mathbb{R}^ n) \] in terms of the potential. This condition was introduced by Palais and Smale in order to find critical points of the action integral. The result of the paper is a negative result: the Palais-Smale condition is an unstable property, i.e., near any bounded potential there exists a potential which does not satisfy the Palais-Smale condition at every large level. The same result is proven in the time-dependent case.