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(uk) → a and G′(uk) → 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.
Keywords
critical point theory, linking, variational methods, saddle
point theory, sandwich pairs, semilinear partial
differential equations, critical sequences
