10 0. MORSE THEORY AND VARIATIONAL PROBLEMS It was also shown in Perera [94] that the assumptions that B is bounded and Φ is bounded from below on bounded sets can be relaxed as follows see also Schechter [118]. Theorem 0.26. If A homologically links B in dimension k, Φ| A ď a ă Φ| B where a is a regular value, and Φ is bounded from below on a set C Ą B such that the inclusion-induced homomorphism r k pW zCq Ñ r k pW zBq is trivial, then Φ has two critical points u1 and u2 with Φpu1q ą a ą Φpu2q, Ck`1pΦ,u1q 0, CkpΦ,u2q 0. Corollary 0.27. Let W W1 W2, u u1 ` u2 be a direct sum decompo- sition with dim W1 k ă 8. If Φ ď a on u1 P W1 : }u1} R ( for some R ą 0, Φ ą a on W2, where a is a regular value, and Φ is bounded from below on tv ` u2 : t ě 0, u2 P W2 for some v P W1z t0u, then Φ has two critical points u1 and u2 with Φpu1q ą a ą Φpu2q, CkpΦ,u1q 0, Ck´1pΦ,u2q 0. The following theorem of Perera and Schechter [103] gives a critical point with a nontrivial critical group in a saddle point theorem with nonstandard geometrical assumptions that do not involve a finite dimensional closed loop see also Perera and Schechter [102] and Lancelotti [63]. Theorem 0.28. Let W W1 W2, u u1 ` u2 be a direct sum decompo- sition with dim W1 k ă 8. If Φ is bounded from above on W1 and from below on W2, then Φ has a critical point u1 with inf ΦpW2q ď Φpu1q ď sup ΦpW1q, CkpΦ,u1q 0. 0.6. Local Linking In many applications Φ has the trivial critical point u 0 and we are interested in finding others. The notion of local linking was introduced by Li and Liu [72, 66], who used it to obtain nontrivial critical points under various assumptions on the behavior of Φ at infinity see also Brezis and Nirenberg [21] and Li and Willem [67]. Definition 0.29. Assume that the origin is a critical point of Φ with Φp0q 0. We say that Φ has a local linking near the origin if there is a direct sum decomposition W W1 W2, u u1 ` u2 with W1 finite dimensional such that $ & % Φpu1q ď 0, u1 P W1, }u1} ď r Φpu2q ą 0, u2 P W2, 0 ă }u2} ď r for sufficiently small r ą 0. Liu [71] showed that this yields a nontrivial critical group at the origin. Theorem 0.30. If Φ has a local linking near the origin with dim W1 k and the origin is an isolated critical point, then CkpΦ, 0q 0.
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