Infinite-dimensional sandwich pairs

For many equations arising in the physical sciences, the solutions are critical points of functionals. This has led to interest in finding critical points of such functionals. If a functional G is semibounded, one can find Palais-Smale (PS) sequences G(u(k)) --> a and G'(u(k)) --> 0. These sequences produce critical points if they have convergent subsequences (that is, if G satisfies the PS condition). However, there is no clear method of finding critical points of functionals that are not semibounded. In this paper we find pairs of sets having the property that functionals bounded from below on one set and bounded from above on the other have PS sequences. We can allow both sets to be infinite-dimensional if we make a slight additional smoothness requirement on the functional. This allows us to solve systems of equations that could not be solved before.