On the Mountain Pass Lemma. (English) Zbl 0604.58015

Chang, Kung-Ching

Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 203-208 (1986).

[For the entire collection see Zbl 0595.00009.]
This is an expository paper on generalizations of the Mountain Pass Lemma (MPL). It extends the MPL in two aspects: (a) from a Banach space to a closed convex subset; (b) from the strong separation condition of values of a function to a weaker one. Theorem. Suppose that a \(C^ 1\) function f satisfies the Palais-Smale condition with respect to a closed convex set C of a Banach space, and that \(\exists \alpha \in R^ 1\) such that \[ Sup\{f(x)| \quad x\in \partial Q\}\leq \alpha,\quad Sup\{f(x)| \quad x\in Q\}<\infty,\quad f(x)>\alpha,\quad \forall x\in S, \] where Q and S are two closed subsets of C, \(\partial Q\) and S link with respect to C. Then one of the three possibilities occurs; (1) \(\alpha\) is an accumulation point of critical values. (2) \(\alpha\) is a critical value with uncountable critical points. (3) \(c=\inf_{A\in F}\text{Sup}_{x\in A}f(x)\) is a critical value, where \(F=\{A=\phi (Q)|\phi\in C(Q,C)\), with \(\phi |_{\partial Q}=id|_{\partial Q}\}.\)
Three applications on multiple solutions of variational inequalities, semilinear elliptic BVP, and minimal surfaces are presented. The first includes some new results and for the last two refer the author [Sci. Sin., Ser. A 26, 1241-1255 (1983; Zbl 0544.35044); 1256-1265 (1983; Zbl 0544.35045)] and the author and J. Eells [Acta Math. Sin., New Ser. 2, 233-247 (1986)].

MSC:

58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E35	Variational inequalities (global problems) in infinite-dimensional spaces
58E12	Variational problems concerning minimal surfaces (problems in two independent variables)

Keywords:

Mountain Pass Lemma; Palais-Smale condition; closed convex set; link; variational inequalities

Citations:

Zbl 0595.00009; Zbl 0544.35044; Zbl 0544.35045