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 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
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.
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