Sandwich pairs in critical point theory

Schechter, Martin

doi:10.1090/S0002-9947-08-04470-X

Sandwich pairs in critical point theory
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Trans. Amer. Math. Soc. 360 (2008), 2811-2823 Request permission

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 \[ G(u_k) \to a,\quad G’(u_k)\to 0. \] These sequences produce critical points if they have convergent subsequences (i.e., if $G$ satisfies the PS condition). However, there is no clear method of finding critical points of functionals which are not semibounded. The concept of linking was developed to produce Palais-Smale (PS) sequences for $C^1$ functionals $G$ that separate linking sets. In the present paper we discuss the situation in which one cannot find linking sets that separate the functional. We introduce a new class of subsets that accomplishes the same results under weaker conditions. We then provide criteria for determining such subsets. Examples and applications are given.

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