8 0. MORSE THEORY AND VARIATIONAL PROBLEMS
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
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
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  and the saddle point and
linking theorems of Rabinowitz [110, 109], respectively. See also Ahmad,
Lazer, and Paul , Castro and Lazer , Benci and Rabinowitz , Ni
, Chang , Qi , and Ghoussoub . 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 , Solimini ,
Ghoussoub , Ramos and Sanchez , and others. However, the notion
of homological linking introduced by Benci [17, 18] and Liu  is better
suited for obtaining critical points with nontrivial critical groups.