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Second order information of Palais-Smale sequences in the mountain pass theorem. (English) Zbl 0797.49036

The authors consider a \(C^2\) functional \(\varphi\) defined on a Hilbert space \(H\) such that the conditions of the Mountain Pass theorem are satisfied. They construct a Palais-Smale sequence \((x_n)\) such that: If \(E\subset H\) is a subspace and \(\langle\varphi''(x_n)u,u\rangle < - \frac{1}{n}\| u\|^ 2\) for every \(u\in E\), then \(\dim E\leq 1\). If the Hessian of \(\varphi\) is non-degenerate, then any cluster point for \((x_n)\) is a critical point of Morse index at most one.
Reviewer: A. Leaci (Lecce)

MSC:

49Q99 Manifolds and measure-geometric topics
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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