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In a recent paper (9) we presented a Galerkin-type Conley index theory for certain classes of innite-dimension al ODEs without the uniqueness property of the Cauchy problem. In this paper we show how to apply this theory to strongly indenite elliptic systems. More specically , we study the elliptic system (A1) u = @vH(u;v;x) in , v = @uH(u;v;x) in , u = 0; v = 0 in @ , on a smooth bounded domain inR N for \ "-type HamiltoniansH of classC 2 satisfying subcritical growth assumptions on their rst order derivatives. As shown by Angenent and van der Vorst in (1), the solutions of (A1) are equilibria of an abstract ordinary dieren tial equation (A2) _

Conley index in Hilbert spaces and a problem of Angenent and van der Vorst