6

Sandwich pairs

**6.1 Introduction**

The notion of sandwich pairs is a useful tool for finding critical points of a functional. It was introduced by Schechter [145, 146] and was based on the sandwich theorem for complementing subspaces by Schechter [137, 138] and Silva [150].

**Definition 6.1.1** We say that a pair of subsets *A, B* of a Banach space *E* is a sandwich pair if every *G* ∈ *C*^{1} (*E*, ℝ) satisfying

and (PS)_{c} for all *c* ∈ [*b*, *a*] has a critical point *u* with *b* ≤ *G*(*u*) ≤ *a*.

**Example 6.1.2** If *E* = *N* ⊕ *M* is a direct sum decomposition with *N* nontrivial and finite dimensional, then *N, M* form a sandwich pair. In fact, we will see in Section 6.3 that this ...

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