Extension of mountain pass lemma. (English) Zbl 0655.58007

Qi, Guijie

Kexue Tongbao, Foreign Lang. Ed. 32, 798-801 (1987).

Suppose that E is a real Banach space and f: \(E\to {\mathbb{R}}\) is a continuously differentiable functional satisfying the Palais-Smale condition. A mountain pass lemma gives the existence of critical points \(x\in E\), i.e. \(f'(x)=0\), provided f satisfies certain geometric conditions. Classical are the restraints given by [A. Ambrosetti and P. H. Rabinowitz, J. Funct. Anal. 14, 349-381 (1973; Zbl 0273.49063)]. They supposed \(f(0)<a=\inf_{x\in S_ R}f(x),\) \(S_ R=\{x\in E| \quad \| x\| =R\}\) and the existence of x’\(\in E\) with \(\| x'\| >R\) and \(f(x')<a\). Then a critical point \(y\in E\), \(y\neq 0\), \(y\neq x'\) exists. A number of generalizations were given by [P. Pucci and J. Serrin, J. Funct. Anal. 59, 185-210 (1984; Zbl 0564.58012)]. The present author gives a generalization to the case where \(f(0)=a=f(x')\).

Reviewer: G.Warnecke

MSC:

58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30	Variational principles in infinite-dimensional spaces
58C25	Differentiable maps on manifolds

Keywords:

Palais-Smale condition; mountain pass lemma; critical points