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Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. (English) Zbl 0872.58018
The paper is concerned with Morse theory for asymptotically quadratic functionals on a Hilbert space X. Let x 0 be an isolated critical point and fC 2 (X,𝐑). Then x 0 may be classified by its critical groups c k (f,x 0 ):=H k ({fc},{fc}{x 0 }), where c=f(x 0 ). In this paper the notion of critical group at infinity is introduced and it is shown that such groups have similar properties to those at finite points. In particular, if f '' () is nondegenerate and has Morse index μ, then c k (f,)0 if and only if k=μ. It is known that if f '' (x 0 ) is degenerate and f satisfies the so-called local linking condition at x 0 , then c k (f,x 0 )0 for some k. A similar result is shown to hold for c k (f,). Moreover, a new “angle condition” is introduced under which the c k ’s behave as in the case of nondegenerate critical point. Applications of the above results are given to the Dirichlet problem -Δu=p(x,u) in Ω, u=0 on Ω, where Ω is a bounded domain in 𝐑 N , p(x,0)=0 and p is asymptotically linear. It is shown that under different conditions c k (f,0)c k (f,) for some k, and therefore this problem has a nontrivial solution. Particular attention is paid to the resonant case.

MSC:
58E05Abstract critical point theory
34B15Nonlinear boundary value problems for ODE
35J65Nonlinear boundary value problems for linear elliptic equations