# zbMATH — the first resource for mathematics

Extension of mountain pass lemma. (English) Zbl 0655.58007
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