AN OVERVIEW xix submanifolds of M, and the positive and negative eigenvalues are given by the critical values of Ψ˘puq 1 J puq , u P M˘, respectively. Let F ˘ denote the class of symmetric subsets of M˘, 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 again 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 ipM`zpΨ`q λ ` k`1 q k (resp. ippΨ´q λ´ k q ipM´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. This will allow us to extend our existence and multiplicity theory for a single equation to systems. For example, suppose F px, 0q 0, so that the system (15) has the trivial solution upxq 0. Assume that F px, uq λ J px, uq ` Gpx, uq where λ is not an eigenvalue of (16) and |Gpx, uq| ď C ÿm i“1 |ui|si @px, uq P Ω ˆ Rm for some si P ppi, p˚q. i Further assume the following superlinearity condition: there are μi ą pi, i 1,...,m such that m ÿ i“1 ˆ 1 pi ´ 1 μi ˙ ui BF Bui is bounded from below and 0 ă F px, uq ď m ÿ i“1 ui μi BF Bui px, uq @x P Ω, |u| large. We will obtain a nontrivial solution of (15) under these assumptions in Sections 10.2 and 10.3.
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