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 W 1, p 0 pΩq 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 λ` k :“ inf M PF ` ipM qěk sup uPM Ψ`puq, λ´ k :“ sup M PF ´ ipM qěk inf uPM Ψ´puq. We will show that λ` k Õ `8 and λ´ k Œ ´8 are sequences of positive and negative eigenvalues, respectively, and if λ` k ă λ` k`1 (resp. λ´ k`1 ă λ´), k then ippΨ`qλk ` q “ ipS ` zpΨ`q λ` k`1 q “ k (resp. ippΨ´q λ ´ k q “ ipS ´ zpΨ´qλk`1 ´ q “ k). In particular, if λ` k ă λ ă λ` k`1 or λ´ k`1 ă λ ă λ´, k then CkpΦλ, 0q ‰ 0. Finally we will present an extension of our theory to anisotropic p-Laplacian systems of the form (15) $ ’ ’ ´Δp i 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
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