xviii AN OVERVIEW for some rij P p1, 1 ` p1 ´ 1{p˚q i p˚q. j Weak solutions of this system are the critical points of the functional Φpuq “ Ipuq ´ ż Ω F px, uq, u P W “ W 1, p1 0 pΩq ˆ ¨ ¨ ¨ ˆ W 1, pm 0 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) $ ’ ’ ´Δp i 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

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2010 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.