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

H kpW zCq Ñ

r

H kpW 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.