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A mountain pass theorem. (English) Zbl 0585.58006
In this paper we establish a new version of the well-known theorem of Ambrosetti and Rabinowitz on the existence of critical points for functionals \(I: X\to {\mathbb{R}}\) of class \(C^ 1\) on a real Banach space X. As usual, a compactness condition of Palais-Smale type is assumed throughout, including a version particularly suited to the periodic case. Roughly speaking, when X is finite dimensional, we show that the mountain pass theorem continues to hold for a mountain of zero altitude. For the infinite dimensional case we obtain the same extension provided the mountain has non-zero thickness. We then derive two corollaries of independent interest. The first one states that if I admits two local minima, then it possesses a third critical point. The second says that if I is v-periodic and e is a local minimum for I, then there exists a critical point \(x\neq e+kv\), \(k=0,\pm 1,\pm 2,... \). As an immediate consequence of this last corollary we obtain the existence of a second independent solution for the forced pendulum equation.

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text: DOI
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