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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
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