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The topological degree at a critical point of mountain-pass type. (English) Zbl 0608.58013

Nonlinear functional analysis and its applications, Proc. Summer. Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/1, 501-509 (1986).
[For the entire collection see Zbl 0583.00018.]
Given a real Banach space E and a \(C^ 1\)-functional \(\Phi\) : \(E\to R\) we define \({\dot \Phi}^ d=\{x\in E|\Phi (x)<d\}\) for a real number d. We say a critical point \(x_ 0\) of \(\Phi\) is of mountain pass type if for every open neighborhood U of \(x_ 0\) the set \(U\cap {\dot \Phi}^ d\) is nonempty and not path connected where \(\Phi (x_ 0)=d\). Under the usual hypotheses of the Ambrosetti-Rabinowitz mountain pass theorem we show the existence of a critical point of mountain pass type. Moreover if \(\Phi\) is a \(C^ 2\)-functional on a real Hilbert space with a gradient of the form identity - compact such that the first eigenvalue \(\lambda_ 1\) of \(\Phi ''(x_ 0)\) is simple provided \(\lambda_ 1=0\), then the Leray Schauder degree at an isolated critical point of mountain pass type is -1. Such \(C^ 2\)-functionals arise in the study of second order elliptic problems.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J05 Equations involving nonlinear operators (general)
34G20 Nonlinear differential equations in abstract spaces

Citations:

Zbl 0583.00018