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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.

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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[1] Ambrosetti, A; Rabinowitz, P.H, Dual variational methods in critical point theory and applications, J. funct. anal, 14, 349-381, (1973) · Zbl 0273.49063
[2] Brezis, H; Coron, J.M; Nirenberg, L, Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz, Comm. pure appl. math, 33, 667-684, (1980) · Zbl 0484.35057
[3] Clark, D.C, A variant of the Lusternik-schnirelman theory, Indiana univ. math. J, 22, 65-74, (1972) · Zbl 0228.58006
[4] \scJ. Mawhin and M. Willem, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation, to appear. · Zbl 0563.34048
[5] Nirenberg, L, Variational and topological methods in nonlinear problems, Bull. amer. math. soc, 4, 267-302, (1981) · Zbl 0468.47040
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