AN OVERVIEW xix

submanifolds of M, and the positive and negative eigenvalues are given by

the critical values of

Ψ˘puq

“

1

J puq

, u P

M˘,

respectively.

Let F

˘

denote the class of symmetric subsets of

M˘,

respectively, and

set

λk

`

:“ inf

M PF

`

ipM qěk

sup

uPM

Ψ`puq,

λk

´

:“ sup

M PF

´

ipM qěk

inf

uPM

Ψ´puq.

We will again show that λk

`

Õ `8 and λk

´

Œ ´8 are sequences of positive

and negative eigenvalues, respectively, and if λk

`

ă λk`1

`

(resp. λk`1

´

ă λk

´),

then

ippΨ`qλk

`

q “

ipM`zpΨ`qλ`

k`1

q “ k

(resp.

ippΨ´qλ´

k

q “

ipM´zpΨ´qλk`1

´

q “ k),

in particular, if λk

`

ă λ ă λk`1

`

or λk`1

´

ă λ ă λk

´,

then

CkpΦλ,

0q ‰ 0.

This will allow us to extend our existence and multiplicity theory for a single

equation to systems.

For example, suppose F px, 0q ” 0, so that the system (15) has the trivial

solution upxq ” 0. Assume that

F px, uq “ λ J px, uq ` Gpx, uq

where λ is not an eigenvalue of (16) and

|Gpx, uq| ď C

ÿm

i“1

|ui|si

@px, uq P Ω ˆ

Rm

for some si P ppi, pi

˚q.

Further assume the following superlinearity condition:

there are μi ą pi, i “ 1,...,m such that

m ÿ

i“1

ˆ

1

pi

´

1

μi

˙

ui

BF

Bui

is bounded

from below and

0 ă F px, uq ď

m

ÿ

i“1

ui

μi

BF

Bui

px, uq @x P Ω, |u| large.

We will obtain a nontrivial solution of (15) under these assumptions in

Sections 10.2 and 10.3.