AN OVERVIEW xvii
where the weight function V P
L8pΩq
changes sign. Here the eigenfunctions
are the critical points of the functional
Φλpuq “
ż
Ω
|∇u|p
´ λ V pxq
|u|p,
u P W0
1, ppΩq
and the positive and negative eigenvalues are the critical values of
Ψ˘puq
“
1
J puq
, u P S
˘,
respectively, where
J puq “
ż
Ω
V pxq
|u|p
and
S
˘
“ u P S : J puq ż 0
(
.
Let F
˘
denote the class of symmetric subsets of S
˘,
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 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 “ ipS
`zpΨ`qλ`
k`1
q “ k
(resp.
ippΨ´qλk
´
q “ ipS
´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.
Finally we will present an extension of our theory to anisotropic
p-Laplacian systems of the form
(15)
$
’
&
’
%
´Δpi ui “
BF
Bui
px, uq in Ω
ui “ 0 on BΩ,
i “ 1,...,m
where each pi P p1, 8q, u “ pu1, . . . , umq, and F P
C1pΩ
ˆ
Rmq
satisfies the
subcritical growth conditions
ˇ
ˇ
ˇ
ˇ
BF
Bui
px, uqˇ
ˇ
ˇ
ˇ
ď C
˜
ÿm
j“1
|uj
|rij ´1
` 1
¸
@px, uq P Ω ˆ
Rm,
i “ 1,...,m
