## 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

Zbl 0583.00018