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A variant mountain pass lemma. (English) Zbl 0544.35044
The Palais-Smale condition depends on the topologies of spaces. The following retractable property is defined to suit for a wider class of spaces. Let X be a Banach space, and let f be a $$C^ 1$$-function defined on it, $$c\in R^ 1$$, we say that f possesses the retractable property with respect to c, if for each $$b>c$$ such that $$f^{-1}(c,b]\cap K=\emptyset$$, $$f_ c$$ is a retraction of $$f_ b$$ (where $$f_{\alpha}$$ is the level set). We extend the mountain pass lemma in two aspects: (1) Replacing the P. S. condition by the retractable property. (2) Dropping out the strictness of the mountain. Namely, Theorem: Suppose that $$f\in C^ 1(X,R^ 1)$$ possesses the retractable property with respect to $$c=f(\theta)$$. Assume that $$\theta$$ is a local minimum of f, and that there is an $$x_ 0\in X$$ such that $$f(x_ 0)=f(\theta)$$; then f has at least one nontrivial critical point. [A more general formulation was announced in the Am. Math. Soc., Notices 26, 805-49-21 (1983).]
The retractable property was verified on the space $$C^ 1_ 0$$ for certain functionals occurred in 2nd order elliptic BVP. A general theorem which combines the sub- and super-solution method with the above M. P. lemma was given. Further applications were obtained in the article reviewed below.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces