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Computations of critical groups at a degenerate critical point for strongly indefinite functionals. (English) Zbl 0982.58009

The paper is concerned with a Morse theory for a class of strongly indefinite asymptotically linear functionals of the form \(f(x) = {1\over 2}\langle Ax,x\rangle + G(x)\), where \(G'\) is compact, \(G'(0)=0\), \(A\) is bounded linear, 0 is an isolated point of finite multiplicity in the spectrum of \(A\) and \(\langle Ax,x\rangle\) is positive- and negative definite on subspaces of infinite dimension. It is shown that under appropriate conditions on the linearization of \(f'\) at 0 the critical groups \(C_{q}(f,0)\) are nonzero only for one \(q\) also in the degenerate case (i.e., when the nullspace of \(f''(0)\neq\{0\}\)). A similar result is shown to hold for the critical groups of \(f\) at infinity.
Finally, the author proves using the Morse inequalities that if \(C_{q}(f,0)\) and \(C_{q}(f,\infty)\) are nonzero for different \(q\)’s, then \(f\) has a critical point \(x_{0}\neq 0\). Moreover, if \(x_{0}\) is nondegenerate, there is a third critical point.
The Morse theory used in this paper is the one developed by the author in her thesis and based on Galerkin-type approximations.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:

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