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A strong form of the mountain pass theorem and application. (English) Zbl 0678.58011
Nonlinear diffusion equations and their equilibrium states I, Proc. Microprogram, Berkeley/Calif. 1986, Publ., Math. Sci. Res. Inst. 12, 341-350 (1988).
[For the entire collection see Zbl 0643.00015.]
A fine structure of the functional near to a mountain pass point, initially studied by Ambrosetti and Rabinowitz, is characterized. Let $$d=\inf \{Sup f(| \gamma |) |$$ $$\gamma$$ is a path connecting $$e_ 0$$ and $$e_ 1\} > \text{Max}\{f(e_ 0),f(e_ 1)\}$$. Assume that the $$(PS)_ d$$ condition for f holds, then there exists a critical point $$u_ 0\in f^{-1}(d)$$ such that (1) $$u_ 0\in \bar B$$, where B is the path component of $$e_ 0$$ in $$\overset\circ f_ d$$, and (2) either for every open neighborhood U of $$u_ 0$$, $$U\cap \overset\circ f_ d\neq \emptyset$$ is not path connected, or $$u_0$$ is the limit of a sequence of local minima, but not a local minimum.
[Related results, cf. P. Pucci and J. Serrin, Trans. Am. Math. Soc. 299, 115-132 (1987; Zbl 0611.58019), J. Differ. Equations 60, 142-149 (1985; Zbl 0585.58006).]
Reviewer: K.Chang

##### MSC:
 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 35J60 Nonlinear elliptic equations