8 0. MORSE THEORY AND VARIATIONAL PROBLEMS 0.5. Linking The notion of homotopical linking is useful for obtaining critical points via the minimax principle. Definition 0.17. Let A be a closed proper subset of a topological space X, g P CpA, W q such that gpAq is closed, B a nonempty closed subset of W such that distpgpAq,Bq ą 0, and Γ “ γ P CpX, W q : γpXq is closed, γ| A “ g ( . We say that pA, gq homotopically links B with respect to X if γpXq X B ‰ H @γ P Γ. When g : A Ă W is the inclusion and X “ tu : u P A, t P r0, 1s ( , we simply say that A homotopically links B. Some standard examples of homotopical linking are the following. Example 0.18. If u0 P W , U is a bounded neighborhood of u0, and u1 R U, then A “ tu0, u1u homotopically links B “ BU . Example 0.19. If W “ W1 ‘W2, u “ u1 `u2 is a direct sum decomposition of W with W1 nontrivial and finite dimensional, then A “ u1 P W1 : }u1} “ R ( homotopically links B “ W2 for any R ą 0. Example 0.20. If W “ W1 ‘W2, u “ u1 `u2 is a direct sum decomposition with W1 finite dimensional and v P W2 with }v} “ 1, then A “ u1 P W1 : }u1} ď R ( Y u “ u1 ` tv : u1 P W1, t ě 0, }u} “ R ( homotopically links B “ u2 P W2 : }u2} “ r for any 0 ă r ă R. Theorem 0.21. If pA, gq homotopically links B with respect to X, c :“ inf γPΓ sup uPγpXq Φpuq is finite, a :“ sup ΦpgpAqq ď inf ΦpBq “: b, and Φ satisfies pCqc, then c ě b and is a critical value of Φ. If c “ b, then Φ has a critical point with critical value c on B. Many authors have contributed to this result. The special cases that correspond to Examples 0.18, 0.19, and 0.20 are the well-known mountain pass lemma of Ambrosetti and Rabinowitz [7] and the saddle point and linking theorems of Rabinowitz [110, 109], respectively. See also Ahmad, Lazer, and Paul [3], Castro and Lazer [24], Benci and Rabinowitz [20], Ni [87], Chang [28], Qi [107], and Ghoussoub [52]. The version given here can be found in Section 3.6. Morse index estimates for a critical point produced by a homotopi- cal linking have been obtained by Lazer and Solimini [65], Solimini [125], Ghoussoub [52], Ramos and Sanchez [112], and others. However, the notion of homological linking introduced by Benci [17, 18] and Liu [71] is better suited for obtaining critical points with nontrivial critical groups.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.