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.