xviii AN OVERVIEW

for some rij P p1, 1 ` p1 ´ 1{pi

˚q

pj

˚q.

Weak solutions of this system are the

critical points of the functional

Φpuq “ Ipuq ´

ż

Ω

F px, uq, u P W “ W0

1, p1

pΩq ˆ ¨ ¨ ¨ ˆ W0

1, pm

pΩq

where

Ipuq “

m ÿ

i“1

1

pi

ż

Ω

|∇ui|pi

.

Unlike in the scalar case, here I is not homogeneous except when p1 “

¨ ¨ ¨ “ pm. However, it still has the following weaker property. Define a

continuous flow on W by

R ˆ W Ñ W, pα, uq Þ Ñ uα :“

p|α|1{p1 ´1

α u1,...,

|α|1{pm ´1

α umq.

Then

Ipuαq “ |α| Ipuq @α P R, u P W.

This suggests that the appropriate class of eigenvalue problems to study

here are of the form

(16)

$

’

&

’

%

´Δpi ui “ λ

BJ

Bui

px, uq in Ω

ui “ 0 on BΩ,

i “ 1,...,m

where J P

C1pΩ

ˆ

Rmq

satisfies

(17) J px, uαq “ |α| J px, uq @α P R, px, uq P Ω ˆ

Rm.

For example,

J px, uq “ V pxq

|u1|r1

¨ ¨ ¨

|um|rm

where ri P p1, piq with r1{p1 ` ¨ ¨ ¨ ` rm{pm “ 1 and V P

L8pΩq.

Note that

(17) implies that if u is an eigenvector associated with λ, then so is uα for

any α ‰ 0.

The eigenfunctions of problem (16) are the critical points of the func-

tional

Φλpuq “ Ipuq ´ λ J puq, u P W

where

J puq “

ż

Ω

J px, uq.

Let

M “ u P W : Ipuq “ 1

(

and suppose that

M˘

“ u P M : J puq ż 0

(

‰ H.

Then M Ă W z t0u is a bounded complete symmetric

C1-Finsler

manifold

radially homeomorphic to the unit sphere in W ,

M˘

are symmetric open