Abstract
Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice, the solutions are critical points. In searching for such points, there is a distinct advantage if the functional G is semibounded. In this case one can find a PalaisβSmale (PS) sequence
G (u~k~) β c, G β²(u~k~) β 0
or even a Cerami sequence
G (u~k~) β c, (1 + βu~k~ β)G β²(u~k~) β 0.
These sequences produce critical points if they have convergent subsequences. However, there is no clear method of finding critical points of functionals which are not semibounded. Linking subsets do provide such a method. They can produce a PS sequence provided they separate the functional. In previous papers we have shown that there are pairs of subsets that can produce Ceramiβlike sequences even though they do not separate the functional. We call such sets sandwich pairs. All that is required is that the functional be bounded from above on one ofthe sets and bounded from below on the other, with no relationship needed between the bounds. This provides a distinct advantage in applications. The present paper discusses the situation in which one cannot find linking subsets which separate the functional or sandwich pairs for which the functional is bounded below on one set and bounded above on the other. We develop a method which can deal with such situations. We apply the method to problems in partial differential equations (Β© 2010 WILEYβVCH Verlag GmbH & Co. KGaA, Weinheim)