Introduction to Minimax Methods

  • Marino Badiale
  • Enrico Serra
Part of the Universitext book series (UTX)


This chapter is an introduction to a broad class of methods that have been shown to be extremely useful in a variety of contexts.

We confine ourselves to the simplest cases, but we try to motivate the ideas involved in the construction of the main tools, so that the interested reader can turn to the study of more complex problems with a minimum of background.

In the preceding chapters we have mainly looked for critical points of a functional as minimum points, either on the whole space, or suitably restricting the functional to sets where minima could be shown to exist. In this chapter, on the contrary, we concentrate on the search of critical points that are not global minima, for example saddle points. The procedures to do this, called minimax methods, are quite elaborate, and we introduce the main steps gradually.


Weak Solution Smale Condition Critical Point Theory Smale Sequence Mountain Pass Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    S. Ahmad, A.C. Lazer, J.L. Paul, Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J. 25(10), 933–944 (1976) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Ambrosetti, A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies in Advanced Mathematics, vol. 104 (Cambridge University Press, Cambridge, 2007) MATHCrossRefGoogle Scholar
  3. 4.
    A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics, vol. 34 (Cambridge University Press, Cambridge, 1995). Corrected reprint of the 1993 original MATHGoogle Scholar
  4. 5.
    A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) MATHCrossRefMathSciNetGoogle Scholar
  5. 7.
    P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7(9), 981–1012 (1983) MATHCrossRefMathSciNetGoogle Scholar
  6. 8.
    T. Bartsch, Z.-Q. Wang, M. Willem, The Dirichlet problem for superlinear elliptic equations, in Stationary Partial Differential Equations. Handbook of Differential Equations, vol. II (Elsevier/North-Holland, Amsterdam, 2005), pp. 1–55 CrossRefGoogle Scholar
  7. 11.
    H. Brezis, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise (Masson, Paris, 1983) MATHGoogle Scholar
  8. 17.
    K. Deimling, Nonlinear Functional Analysis (Springer, Berlin, 1985) MATHGoogle Scholar
  9. 18.
    L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998) MATHGoogle Scholar
  10. 26.
    O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & Applications, vol. 13 (Springer, Paris, 1993) MATHGoogle Scholar
  11. 28.
    E.M. Landesman, A.C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 19, 609–623 (1969/1970) MathSciNetGoogle Scholar
  12. 35.
    J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74 (Springer, New York, 1989) MATHGoogle Scholar
  13. 37.
    W.-M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications. Indiana Univ. Math. J. 31(6), 801–807 (1982) MATHCrossRefMathSciNetGoogle Scholar
  14. 38.
    R. S Palais, The principle of symmetric criticality. Commun. Math. Phys. 69(1), 19–30 (1979) MATHCrossRefGoogle Scholar
  15. 40.
    R.S. Palais, S. Smale, A generalized Morse theory. Bull. Am. Math. Soc. 70, 165–172 (1964) MATHCrossRefMathSciNetGoogle Scholar
  16. 42.
    P.H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, in Nonlinear Analysis (Collection of Papers in Honor of Erich H. Rothe) (Academic Press, New York, 1978), pp. 161–177 Google Scholar
  17. 43.
    P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 65 (American Mathematical Society, Providence, 1986) Google Scholar
  18. 44.
    S. Solimini, On the solvability of some elliptic partial differential equations with the linear part at resonance. J. Math. Anal. Appl. 117(1), 138–152 (1986) MATHCrossRefMathSciNetGoogle Scholar
  19. 45.
    M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge [Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics], vol. 34 (Springer, Berlin, 2008) MATHGoogle Scholar
  20. 46.
    M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187(4), 511–517 (1984) MATHCrossRefMathSciNetGoogle Scholar
  21. 48.
    M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24 (Birkhäuser, Boston, 1996) MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TorinoTorinoItaly

Personalised recommendations