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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.