G(u k ) ’ a = inf G > ’∞.

(1.3)

If this series converges or has a convergent subsequence, we have a minimum. How-

ever, we have no guarantee that a subsequence will converge. Moreover, a minimizing

sequence provides very little structure that can help one ¬nd a convergent subsequence.

Indeed, one can ¬nd simple examples of functionals that are bounded below and have

no minimum. For instance, the function

y(x) = e x

has no minimum on R even though it is bounded from below.

1.3 Palais“Smale sequences

On the other hand, if the functional G is bounded from below, it can be shown that

there is a sequence satisfying

G(u k ) ’ a, G (u k ) ’ 0.

(1.4)

(Actually, one can do better; see below.) If the sequence has a convergent subse-

quence, this will produce a minimum. Such a sequence is called a Palais“Smale (PS)

sequence. The advantage of having a PS sequence instead of a minimizing sequence

is that the additional information

G (u k ) ’ 0

helps one show that a PS sequence has a convergent subsequence more readily than

one can show that a minimizing sequence has a convergent subsequence.

1.4. Cerami sequences 3

1.4 Cerami sequences

Actually, it can be shown that a C 1 -functional G that is bounded from below has a

Cerami sequence, a sequence satisfying

G(u k ) ’ a, (1 + u k )G (u k ) ’ 0

(1.5)

(cf. Theorem 3.21 and its corollary). A Cerami sequence says more. There are

examples for which one can show that a Cerami sequence has a convergent subse-

quence, while one cannot do the same for the corresponding PS sequence.

1.5 Linking sets

When the functional is not semibounded, there is no clear way of obtaining critical

points. In general, one would like to determine when a functional has PS or Cerami

sequences. That is, one would like to ¬nd situations that give one the same advantages

that one has in the case of semibounded functionals.

An idea that has been very successful is to ¬nd appropriate sets that separate the

functional. By this we mean the following:

De¬nition 1.1. Two sets A, B separate the functional G(u) if

a0 := sup G < b0 := inf G.

(1.6)

B

A

We would like to ¬nd sets A and B such that (1.6) will imply

∃ u : G(u) ≥ b0 , G (u) = 0.

(1.7)

This is too much to expect, since even semiboundedness alone does not imply the

existence of a critical point. Consequently, we weaken our requirements and look for

sets A, B such that (1.6) implies the existence of a PS sequence (1.4) with a ≥ b0 . This

leads to

De¬nition 1.2. We shall say that the set A links the set B if (1.6) implies (1.4) with

a ≥ b0 for every C 1 -functional G(u).

Of course, (1.4) is a far cry from (1.7), but if, e.g., the sequence (1.4) has a con-

vergent subsequence, then (1.4) implies (1.7). Whether or not (1.4) implies (1.7) is a

property of the functional G(u). We state this as

De¬nition 1.3. We say that G(u) satis¬es the PS condition if (1.4) always implies

(1.7).

The usual way of verifying this is to show that every sequence satisfying (1.4) has

a convergent subsequence (there are other ways).

4 1. Critical Points of Functionals

All of this leads to

Theorem 1.4. If G satis¬es the PS condition and is separated by a pair of linking sets,

then it has a critical point satisfying (1.7).

This theorem cannot be applied until one identi¬es linking sets and functionals that

satisfy the PS condition. Fortunately, they exist. We shall describe many of the known

linking sets later in the book.

Among other things, we shall discuss when

a0 := sup G ¤ b0 := inf G

(1.8)

B

A

implies the existence of a PS sequence or a Cerami sequence.

If there do not exist linking sets that separate a functional, all is not lost. There

exist sets that produce PS or Cerami sequences when they do not separate a functional.

That is, we shall ¬nd sets such that

’∞ < b0 := inf G, a0 := sup G < ∞

(1.9)

B A

implies the existence of a PS sequence or a Cerami sequence.

1.6 Previous de¬nitions of linking

In the earlier versions of linking, A is a compact set and is the “boundary” of a manifold

S. They say that A links a set B if

A©B =φ

and every continuous map • from S to E that equals the identity on A must satisfy

•(S) © B = φ.

The underlying theorem is

Theorem 1.5. If the set A links the set B and (1.6) holds, then there is a Palais“Smale

sequence satisfying (1.4) with a ≥ b0 .

There are distinct disadvantages of this de¬nition. First, it requires A to be compact.

Second, it requires A to be the “boundary” of a manifold S. Third, linking depends on

the manifold S. Finally, there is no possibility of symmetry in in¬nite-dimensional

spaces, i.e., if A links B, then B cannot link A according to this de¬nition.

Several methods have been used to circumvent these shortcomings and extend the

de¬nition of linking to cover cases when the old de¬nition does not apply. Some of

them are described in Chapter 3. The purpose of this volume is to present a uniform

approach that includes most, if not all, theories of linking. It employs minimax systems

and is presented in Chapter 2.

The outline of this book is as follows. In Chapter 5 we give the proofs of the

theorems of Chapter 2. We use methods of solving differential equations in Banach

1.7. Notes and remarks 5

spaces described in Chapter 4. In Chapter 6 we show that our uniform method is almost

identical to the general De¬nition 1.2 given above. In Chapter 7 we consider the case

when there are no linking sets that separate the functional. We ¬nd pairs of sets (called

sandwich pairs) that produce PS sequences (1.4) when (1.6) fails. In Chapters 8, 9,

11, 14, 16, and 17 we give some applications to partial differential equations of the

methods presented in Chapters 1“7. In Chapters 10, 12, 13, and 15 we are concerned

with the situation when A and B are both in¬nite-dimensional.

1.7 Notes and remarks

For an excellent review of critical point theory and linking, we refer to [96]. Previous

de¬nitions of linking theory are described in [8]. Other de¬nitions of linking will be

discussed in Chapters 2 and 3.