Extensions of the mountain pass theorem.

*(English)*Zbl 0564.58012The paper contains a number of extensions of the mountain pass lemma of A. Ambrosetti and P. H. Rabinowitz [(*) ibid. 14, 349-381 (1973; Zbl 0273.49063)]. The lemma gives sufficient conditions for the existence of critical points of continuously FrĂ©chet differentiable functionals \(I: X\to {\mathbb{R}}\) on a real Banach space X. The hypotheses of the lemma and its variants consist of a compactness condition and geometric restraints on the functional I. It was shown in (*) how the lemma may be applied to prove the existence of weak solutions for differential equations. (See also the survey article by L. Nirenberg [Bull. Am. Math. Soc., New Ser. 4, 267-302 (1981; Zbl 0468.47040)] for an introduction.)

The authors of the paper under review study variants of the geometric restraints on the functional I. At the same time they make statements as to whether one obtains local minima, maxima or saddle points. For example, take \(K_ b=\{x\in X| \quad I(x)=b,\quad I'(x)=0\},\) the set of all critical points with critical value b. If b is the value given in the original mountain pass lemma and X is infinite dimensional, then \(K_ b\) contains at least one saddle point. Finally, the authors give modifications of the above-mentioned results for periodic functionals. In this case one needs an adapted version of the compactness condition. For a different type of extension of the mountain pass lemma and its applications we would like to mention results of M. Struwe [Math. Ann. 261, 399-412 (1982; Zbl 0506.35034); J. Reine Angew. Math. 349, 1-23 (1984; Zbl 0521.49028)]. In these papers the differentiability requirement for the functional I is weakend.

The authors of the paper under review study variants of the geometric restraints on the functional I. At the same time they make statements as to whether one obtains local minima, maxima or saddle points. For example, take \(K_ b=\{x\in X| \quad I(x)=b,\quad I'(x)=0\},\) the set of all critical points with critical value b. If b is the value given in the original mountain pass lemma and X is infinite dimensional, then \(K_ b\) contains at least one saddle point. Finally, the authors give modifications of the above-mentioned results for periodic functionals. In this case one needs an adapted version of the compactness condition. For a different type of extension of the mountain pass lemma and its applications we would like to mention results of M. Struwe [Math. Ann. 261, 399-412 (1982; Zbl 0506.35034); J. Reine Angew. Math. 349, 1-23 (1984; Zbl 0521.49028)]. In these papers the differentiability requirement for the functional I is weakend.

Reviewer: G.Warnecke

##### MSC:

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

57R70 | Critical points and critical submanifolds in differential topology |

49Q99 | Manifolds and measure-geometric topics |

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\textit{P. Pucci} and \textit{J. Serrin}, J. Funct. Anal. 59, 185--210 (1984; Zbl 0564.58012)

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##### References:

[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] | Hofer, H, A note on the topological degree at a critical point of mountain pass type, (), 309-315 · Zbl 0545.58015 |

[5] | Mawhin, J; Willem, M, Variational methods and boundary value problems for vector second order differential equations and applications to the pendulum equation, J. diff. equations, 52, 264-287, (1984) · Zbl 0557.34036 |

[6] | \scP. Pucci and J. Serrin, A mountain pass theorem J. Diff. Equations, in press. · Zbl 0585.58006 |

[7] | Ni, W.M, Some minimax principles and their applications in nonlinear eliptic equations, J. analyse math., 37, 248-278, (1980) |

[8] | Rabinowitz, P.H, Variational methods for nonlinear eigenvalue problems, (), Varenna · Zbl 0212.16504 |

[9] | Rabinowitz, P.H, Some aspects of critical point theory, University of wisconsin, MRC technical report no. 2465, (1983) |

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