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.
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