0.4. MINIMAX PRINCIPLE 7

0.4. Minimax Principle

Minimax principle originated in the work of Ljusternik and Schnirelmann

[76] and is a useful tool for finding critical points of a functional. Note that

the first deformation lemma implies that if c is a regular value and Φ satisfies

pCqc, then the family Dc,

ε

of maps η P Cpr0, 1s ˆ W, W q satisfying

piq ηp0, ¨q “ id

W

,

piiq ηpt, ¨q is a homeomorphism of W for all t P r0, 1s,

piiiq ηpt, ¨q is the identity outside Φc´2ε

c`2ε

for all t P r0, 1s,

pivq Φpηp¨,uqq is nonincreasing for all u P W ,

pvq ηp1,

Φc`εq

Ă

Φc´ε

is nonempty for all suﬃciently small ε ą 0. We say that a family F of

subsets of W is invariant under Dc,

ε

if

M P F, η P Dc,

ε

ùñ ηp1,M q P F.

Theorem 0.14 (Minimax Principle). If F is a family of subsets of W ,

c :“ inf

M PF

sup

uPM

Φpuq

is finite, F is invariant under Dc,

ε

for all suﬃciently small ε ą 0, and Φ

satisfies pCqc, then c is a critical value of Φ.

We say that a family Γ of continuous maps γ from a topological space

X into W is invariant under Dc,

ε

if

γ P Γ, η P Dc,

ε

ùñ ηp1, ¨q ˝ γ P Γ.

Minimax principle is often applied in the following form, which follows by

taking F “ γpXq : γ P Γ

(

in Theorem 0.14.

Theorem 0.15. If Γ is a family of continuous maps γ from a topological

space X into W ,

c :“ inf

γPΓ

sup

uPγpXq

Φpuq

is finite, Γ is invariant under Dc,

ε

for all suﬃciently small ε ą 0, and Φ

satisfies pCqc, then c is a critical value of Φ.

Some references for Theorems 0.14 and 0.15 are Palais [93], Nirenberg

[89], Rabinowitz [111], and Ghoussoub [53].

Minimax methods were introduced in Morse theory by Marino and Prodi

[79]. The following result is due to Liu [71].

Theorem 0.16. If σ P

HkpΦb, Φaq

is a nontrivial singular homology class

where ´8 ă a ă b ď `8 are regular values,

c :“ inf

zPσ

sup

uP|z|

Φpuq

where |z| denotes the support of the singular chain z, Φ satisfies pCqc, and

Kc

is a finite set, then there is a u P

Kc

with CkpΦ,uq ‰ 0.