## On the Mountain Pass Lemma.(English)Zbl 0604.58015

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:

 5.8e+06 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 5.8e+36 Variational inequalities (global problems) in infinite-dimensional spaces 5.8e+13 Variational problems concerning minimal surfaces (problems in two independent variables)

### Citations:

Zbl 0595.00009; Zbl 0544.35044; Zbl 0544.35045