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