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 :“
p|α|1{p1 ´1
α u1,...,
|α|1{pm ´1
α umq.
Then
Ipuαq |α| Ipuq 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 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 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

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 ,

are symmetric open
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