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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.

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