Schechter Minimax Systems and Critical Point Theory (Springer, 2009)

2.3. Linking subsets

9

Theorem 2.6. Let K be a minimax system for a nonempty set A and let G(u) be a C1 functional on E such that

(2.10)		gK = {v K \ A : G(v) ≥ a0} = φ, K K,
and
(2.11)			a = inf sup G < ∞.
			K	K
Assume that for any b > a, K K and flow σ (t) satisfying
(2.12)		G(σ (t)v) < b,		v K , 0 < t ≤ T,
and
(2.13)		G(σ (t)u) < a,		u A, 0 < t ≤ T,
K	K	such that
there is a ˜		˜	σ (t) A σ (T )[Eb K ],
(2.14)
		K
			t [0,T ]
where
(2.15)			Eα = {u E : G(u) ≤ α}.

Then the conclusions of Theorem 2.4 hold.

2.3 Linking subsets

We now show how linking can play a major role in finding critical points.

Definition 2.7. We shall say that a set A in E links a set B E relative to a minimax system K for A if

(2.16)	A ∩ B = φ
and
(2.17)	B ∩ K = φ, K K.

We shall say that A links B [mm] if there is a minimax system K for A such that A links B relative to K.

Theorem 2.8. Let K be a minimax system for a nonempty set A, and assume that there is a subset B of E such that A links B relative to K. Assume that G C1(E, R) satisfies

(2.18)	a	0	:= sup G < b0 :=	inf G
				B
			A

and that the quantity a given by (2.6) is finite. Then, for each positive, nonincreasing, locally Lipschitz continuous function ψ(t) on [0, ∞) satisfying (2.8), there is a sequence {uk } E such that (2.9) holds.

10	2. Minimax Systems

Definition 2.9. A subset A of a Banach space E links a subset B of E if, for every G C1(E, R) bounded on bounded sets and satisfying (2.18) there are a sequence {uk } E and a constant a such that

(2.19)	b0 ≤ a < ∞
and (1.4) holds.

Definition 2.10. A subset A of a Banach space E links a subset B of E strongly if, for every G C1(E, R) bounded on bounded sets and satisfying (2.18) and each positive, nonincreasing, locally Lipschitz continuous function ψ(t) on [0, ∞) satisfying (2.8), there is a sequence {uk } E such that (2.9) and (2.19) hold.

Theorem 2.11. If A links B [mm], then it links B strongly.

We note that Theorem 1.5 is a simple consequence of Theorem 2.4. In fact, we can let K be the collection

K = {ϕ(S) : ϕ C(S, E), ϕ(u) = u, u A}.

It is easily checked that K is a minimax system for A. Moreover, if A links B in the old sense, then it links B [mm]. We can now apply Theorem 2.4.

In the following theorems, we allow a = a0.

Theorem 2.12. Let K be a minimax system for a nonempty set A, and assume that A links a subset B of E relative to K. Let G be a C1-functional satisfying (2.10), (2.11), and

(2.20)	a	0	:= sup G ≤ b0	:=	inf G	.
					B
			A

Assume that for any b > a, K K, and flow σ (t) satisfying (2.12), (2.13) and such that

(2.21)			σ (t) A ∩ B = φ,	t [0, T ],
K		K	such that (2.14) holds. Then the conclusions of Theorem 2.4 hold.
there is a ˜
Moreover, if a0 = a, then there is a sequence satisfying (1.4) and
(2.22)			d(uk , B) → 0,	k → ∞.
Theorem 2.13. Let K be a minimax system for a nonempty set A satisfying
(2.23)			K \ A = φ,	K K.

Let G	C1 satisfy (2.10) and (2.11). Assume that for any K K and flow σ (t)
satisfying (2.12), (2.13), one has S(K ) K, where
(2.24)	S(u) = σ (d(u, A))u, u E.

Assume also that for each K K, there is a ρ > 0 such that G(u) is Lipschitz continuous on

(2.25) Kρ = {u K : d(u, A) ≤ ρ}.

Then the conclusions of Theorem 2.4 hold.

2.4. A variation

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Theorem 2.14. Let K be a minimax system for a nonempty set A, and assume that there is a subset B of E such that A links B relative to K. Assume that for any K K and flow σ (t) satisfying (2.12), (2.13), one has S(K ) K, where S(u) is given by (2.24). Let G be a C1-functional satisfying (2.11) and (2.20). Assume also that for each K K there is a ρ > 0 such that G(u) is Lipschitz continuous on Kρ given by (2.25). Then the conclusions of Theorem 2.4 hold.

Corollary 2.15. Let K be a minimax system for a nonempty set A, and let G C1 satisfy (2.10) and (2.11). Assume that for any b > a, K K, and flow σ (t) satisfying (2.12), (2.13), and (2.21) for

(2.26)	B = {u E\ A : G(u) ≥ a0},
there is a K	K	T	)	E	b. Then the conclusions of Theorem 2.4 hold.
˜ K satisfying	˜ σ (

2.4 A variation

Theorem 2.16. Let K be a minimax system for a nonempty subset A of E, and let G(u) be a C1-functional on E. Define

(2.27)	b := sup	inf G		,
			K
	K K
and assume that b is finite and satisfies
(2.28)	b < b0 :=		inf G	.
			A

Then the conclusions of Theorem 2.4 hold with a replaced by b.

We also have the following. Let A = {0}. Then a minimax system for A can be constructed as follows. K consists of the boundaries of all bounded, open sets ω containing 0. That K is a minimax system for A follows from the fact that 0 ϕ(ω) and ∂ ϕ(ω) = ϕ(∂ ω) for all such sets and ϕ ( A). We have

Theorem 2.17. Let b be defined by (2.27). Assume that G(0) < b < ∞. Then there is a sequence satisfying

(2.29)	G(uk )	→	b, G	(uk )/ψ( uk	)	0.
					→

Theorem 2.18. Assume that b < ∞ and that there is an open set ω0 containing 0 such that

G(0) = inf G.

∂ω0

Then there is a sequence satisfying (2.29).

We also have the counterpart of these theorems for the quantity a given by (2.6).

Theorem 2.19. Assume that −∞ < a < G(0). Then the conclusions of Theorem 2.4 hold.

12	2. Minimax Systems

Theorem 2.20. Assume that a > −∞ and that there is an open set ω0 containing 0 such that

G(0) = sup G.

∂ω0

Then the conclusions of Theorem 2.4 hold.

2.5 Weaker conditions

We now turn to the question as to what happens if some of the hypotheses of Theorem 2.12 do not hold. We are particularly interested in what happens when (2.20) is violated. In this case we let

(2.30)	B := {v B : G(v) < a0}.
Note that	B = φ iff a0 ≤ b0.

We assume that B = φ. Let ψ(t) be a positive, nonincreasing function on [0, ∞) satisfying the hypotheses of Theorem 2.4 and such that

(2.31)	a0 − b0 <		R+α
		α	ψ(t) dt

for some finite R ≤ d := d(B , A), where α = d(0, B ). We assume d > 0. We have

Theorem 2.21. Let G be a C1-functional on E and A, B E be such that A links B [mm] and

(2.32) −∞ < b0, a < ∞.

Assume that for any b > a, K K and flow σ (t) satisfying (2.12), (2.13), (2.21), there

is a ˜	K	satisfying (2.14). Under the hypotheses given above, for each	δ >	0, there
K
is a u E such that
(2.33)		b0 − δ ≤ G(u) ≤ a + δ, G (u) < ψ(d(u, B )).

We can also consider a slightly different version of Theorem 2.21. We consider

the set
(2.34)	A := {u A : G(u) > b0},

and we note that A = φ iff a0 ≤ b0. We assume that ψ satisfies the hypotheses of Theorem 2.4 and

(2.35)	a0 − b0 <		R+β
		β	ψ(t) dt
holds for some finite R ≤ d	:= d( A , B), where β = d(0, A ). Assume that A = φ.
We have

2.6. Some consequences						13
Theorem 2.22. Assume that for any b < b0, K K, and flow σ (t) satisfying
(2.36)		G(σ (t)v) > b,		v K , t > 0,
(2.37)		G(σ (t)v) > b0,		v B, t > 0,
and
(2.38)		σ (t)B ∩ A = φ, t [0, T ],
K	K	such that
there is a ˜		˜
(2.39)			σ (t)B σ (T )[E		b	K ],
		K
			t [0,T ]
where
(2.40)			Eα = {u E : G(u) ≥ α}.
If B links A [mm] and
(2.41)			−∞ < b0,	a < ∞
hold, then, for each δ > 0, there is a u E such that
(2.42)	b0 − δ ≤ G(u) ≤ a + δ,			G (u) < ψ(d(u, A )).

2.6 Some consequences

We now discuss some methods that follow from Theorems 2.21 and 2.22. Let { Ak , Bk }

be a sequence of pairs of subsets of E. Let Kk be a minimax system for Ak . For G C1(E, R), let

(2.43)		a		= sup G,	bk0	inf	G,
			k0			= Bk
				Ak
and
(2.44)				ak = Kinf	sup G.
				Kk K
We assume ak < ∞ for each k. We define
(2.45)	Bk	:= {v Bk : G(v) < ak0},
(2.46)	Ak	:= {u Ak : G(u) > bk0}
(2.47)	d	: d( Ak , B ),			d	: d( A		, Bk ).
	k	=		k	k	=	k

14 2. Minimax Systems

We have

Theorem 2.23. Assume that Ak links Bk [mm] for each k, that

(2.48) dk → ∞ as k → ∞,

and for each k, there is a positive, nonincreasing function ψk (t) on [0, ∞) satisfying the hypotheses of Theorem 2.4 and such that

(2.49)	ak0 − bk0 <	Rk +αk	ψk (t)dt,
		αk

where αk = d(0, Bk ) and Rk ≤ dk . Then, under the hypotheses of Theorem 2.21, there is a sequence {uk } E such that

(2.50)	bk0 − (1/ k) ≤ G(uk ) ≤ ak + (1/ k)
and
(2.51)		G	(u )	≤	ψ (d(u	, B		)).
			k		k k		k

Theorem 2.24. Assume that Bk links Ak [mm] for each k, that

(2.52)

dk → ∞ as k → ∞,

and that, for each k, there is a positive, nonincreasing function ψk (t) on [0, ∞) satisfying the hypotheses of Theorem 2.4 and such that

(2.53)	ak0 − bk0 <	Rk +βk	ψk (t)dt,
		βk

where βk = d(0, Ak ) and Rk ≤ dk . Then, under the hypotheses of Theorem 2.22, there is a sequence {uk } E such that

(2.54)	bk0 − (1/ k) ≤ G(uk ) ≤ ak + (1/ k)
and
(2.55)		G	(uk )	≤	ψk (d(uk , A )).
					k

We combine the proofs of Theorems 2.23 and 2.24.

Proof. For each k, take Rk equal to dk or dk , as the case may be. We may assume that bk0 < ak0 for each k. Otherwise the conclusions of the theorems follow from Theorem 2.4. We can now apply Theorems 2.21 and 2.22 for each k to conclude that there is a uk E such that

bk0 − (1/ k) ≤ G(uk ) ≤ ak + (1/ k),

and either

G (uk ) < ψk (d(uk , Bk ))

or

G (uk ) < ψk (d(uk , Ak )),

as the case may be.

2.7. Notes and remarks

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2.7 Notes and remarks

The concept of a minimax system is related to that of [74] (cf. Section 3.4). The first situation that allowed a0 = a appeared in [75]. The first to consider the case ψ(t) = 1/(1 + |t|) was [39]. The first to consider the general case of arbitrary nonincreasing ψ was [107]. The theorems of Section 2.4 are due to [107]. The theorems of Sections 2.2, 2.3, 2.5, and 2.6 are due to [134].

Chapter 3

Examples of Minimax Systems

3.1 Introduction

In this chapter we present some linking methods which are special cases of linking with respect to minimax systems. We consider three useful methods and show that they provide minimax systems. We then consider examples and applications.

3.2 A method using homeomorphisms

One linking method can be described be the set of all continuous maps =

as follows. Let E be a Banach space and let(t) from E × [0, 1] to E such that

1.(0) = I , the identity map.

2.For each t [0, 1), (t) is a homeomorphism of E onto E and −1(t) C(E × [0, 1), E).

3.(1)E is a single point in E and (t) A converges uniformly to (1)E as t → 1 for each bounded set A E.

4.For each t0 [0, 1) and each bounded set A E,

(3.1)	sup { (t)u + −1(t)u } < ∞.
	0≤t≤t0
	u A
We have

Theorem 3.1. Let G be a C1-functional on E, and let A be a subset of E. Assume that

(3.2)	a	0 := sup G < a :=	0 s 1		G( (s)u) < ∞.
			inf	sup
		A		≤ ≤
				u A

Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞) satisfying (2.8). Then there is a sequence {uk } E such that (2.9) holds. In particular, there is a Cerami sequence satisfying (1.5).

M. Schechter, Minimax Systems and Critical Point Theory, DOI 10.1007/978-0-8176-4902-9_3, © Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2009

18	3. Examples of Minimax Systems
Proof. In this case, we let K be the collection of sets of the form
(3.3)	K = { (t)u : t [0, 1], u A, }.

To show that this is a minimax system, let ϕ be a mapping in ( A), and let (t) be in. Define the mapping

1(t)u = ϕ ◦ (t) ◦ ϕ−1u, u E.

Then 1(t) . Moreover,

1(t)u = ϕ ◦ (t)u, u A.

Consequently,

1(t) A = ϕ({ (t)u : u A}),

showing that the collection K is a minimax system for A. Since (3.2) now becomes (2.7), the result follows.

Definition 3.2. We say that A links B [hm] if A, B are subsets of E such that A∩ B = φ and, for each (t) , there is a t (0, 1] such that (t) A ∩ B = φ.

We have

Corollary 3.3. If A links B [hm], then it links B [mm].

We also have

Theorem 3.4. Let A, B be subsets of E such that A links B [hm], and let G be a C1-functional on E, satisfying

(3.4)	a0 := sup G ≤ b0		:	inf G			.
			=	B
		A
Assume that	=	0≤s≤1
(3.5)			( ( )		u	)
	a :	inf sup G		s

u A

is finite. Let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞) satisfying (2.8). Then there is a sequence {uk } E such that (2.9) holds. In particular, there is a Cerami sequence satisfying (1.5).

Proof. The theorem follows from Theorem 2.12. If

(3.6)		K = { (t)u : t [0, 1], u A}
for some , and σ (t) is any flow, let						0 ≤ s ≤ 21 ,
(3.7)	(s)	=		σ (2T s),
	˜			σ (T ) (2s − 1),		1
	˜					2 < s ≤ 1.
			. Moreover,
It is easily checked that
(3.8)			˜		σ (t) A σ (T )K .
		K

t [0,T ]

Hence, (2.14) is satisfied.

3.3. A method using metric spaces

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3.3 A method using metric spaces

Another general procedure is described in Mawhin–Willem [96] and Brezis–Nirenberg

[28] as follows. One finds a compact metric space and selects a closed subset ofsuch that = φ, = . One then picks a map p C( , E) and defines

A ={ p C( , E) : p = p on },

a = inf max G( p(ξ )).

p A ξ

They assume

(A) For each p A, maxξ G( p(ξ )) is attained at a point in \ . They then prove

Theorem 3.5. Under the above hypotheses, there is a sequence satisfying (1.4).

We shall prove

Theorem 3.6. Under the same hypotheses, let ψ(t) be a positive, nonincreasing, locally Lipschitz continuous function on [0, ∞) satisfying (2.8). Then there is a sequence {uk } E such that (2.9) holds. In particular, there is a Cerami sequence satisfying (1.5).

Proof. If a > a0, this follows from Theorem 2.4. In fact, we can take A = p ( ) and K as the collection of sets

K = { p(ξ ) : ξ , p A}.

If ϕ ( A), then

ϕ( p(ξ )) = ϕ( p (ξ )) = p (ξ ), ξ .

Thus, ϕ( p(ξ )) K. Hence, K is a minimax system for A, and we can apply Theorem 2.4 to come to the desired conclusion.

Now assume that a0 = a. Since each K K is compact, the same is true of Kρ

given by (2.25) for each ρ > 0. Moreover, if σ (t)	C(R+ × E, E) is such that
σ (0) = I, one has S(K ) K, where
S( p(ξ )) = σ (d( p(ξ ), A)) p(ξ ),	ξ .
The result follows from Theorem 2.14.

3.4 A method using homotopy-stable families

Another approach is found in Ghoussoub [74]. The author considers a closed subset A of E, and a collection K of compact subsets of E having the property that if K K and ψ C([0, 1] × E, E) satisfies

ψ(0)u = u, u E,