Title: Strong sandwich pairs

Authors: Martin Schechter

Issue: Volume 57 (2008), Issue 3, 1105-1132

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 of functionals. If a functional $G$ is semibounded, one can find a Palais-Smale (PS) sequence \[ G(u_k) \rightarrow c, \quad G\prime(u_k) \rightarrow 0 \] or even a Cerami sequence \[ G(u_k) \rightarrow c, \quad (1 + \| {u_k} \|G\prime(u_k) \rightarrow 0. \] These sequences produce critical points if they have convergent subsequences. Cerami sequences are superior to PS sequences because they provide more structure that can be used to verify the existence of a convergent subsequence. However, there is no clear method of finding critical points of functionals which are not semibounded. In the present paper we develop a method of finding structured sequences for functionals which are not semibounded. These sequences are essentially as good as Cerami sequences and provide a distinct advantage in applications. We apply the method to several situations.