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 :“ 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) $ ´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 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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