Minimization and Mountain-Pass Theorems

  • Maria do Rosário Grossinho
  • Stepan Agop Tersian
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 52)


In this introductory chapter, we consider the concept on differentiability of mappings in Banach spaces, Fréchet and Gâteaux derivatives, secondorder derivatives and general minimization theorems. Variational principles of Ekeland [Ek1] and Borwein & Preiss [BP] are proved and relations to the minimization problem are given. Deformation lemmata, Palais—Smale conditions and mountain-pass theorems are considered. The deformation approach and ε—variational approach are applied to prove the mountainpass theorem and its various extensions.


Banach Space Convergent Subsequence Critical Point Theory Minimax Theorem Critical Point Theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [APr]
    Ambrosetti, Antonio and Prodi, Giovanni. A Primer of Nonlinear Analysis. Cambridge Univ. Press, 1994.Google Scholar
  2. [ARa]
    Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J. Funct. Anal., 1973;14:349–381.MathSciNetMATHCrossRefGoogle Scholar
  3. [AEk]
    Aubin, Jean-Pierre and Ekeland, Ivar. Applied Nonlinear Analysis. N.Y.: John Wiley & Sons, 1984.MATHGoogle Scholar
  4. [BBF]
    Bartolo P, Benci V, Fortunato D. Abstract critical point theory and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. T.M.A., 1983;7,9:981–1012.MathSciNetCrossRefGoogle Scholar
  5. [Ber]
    Berger, Melvin. Nonlinearity and Functional Analiysis. N.Y.: Academic Press, 1977.MATHGoogle Scholar
  6. [BP]
    Borwein J, Preiss D. A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions. Trans. A.M.S., 1987;303:517–527.MathSciNetMATHCrossRefGoogle Scholar
  7. [BCN]
    Brezis H, Coron JM, Nirenberg L. Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz. Comm. Pure and Appl. Math., 1980;33:667–689.MathSciNetMATHCrossRefGoogle Scholar
  8. [BN]
    Brezis H, Nirenberg L. Remarks on finding critical points. Comm. Pure and Appl. Math., 1991;XLIV:939–963.MathSciNetCrossRefGoogle Scholar
  9. [Car]
    Cartan, Henri. Calcul différentiel, Formes différentielles. Paris: Hermann,1967.Google Scholar
  10. [Ce]
    Cerami G. Un criterio di esistenza per punti critici su varieta illimate. Rend. Acad. Sci. Let. Ist. Lombardo 1978;112:332–336.MathSciNetMATHGoogle Scholar
  11. [Ch1]
    Chang, Kung. Infinite Dimensional Morse Theory and Multiple Solution Problems. Boston, Basel, Berlin: Birkhäuser, 1993.MATHCrossRefGoogle Scholar
  12. [Co]
    Cohn Donald. Measure Theory, Birkhäser, 1980.MATHGoogle Scholar
  13. [DM]
    Degiovanni M, Marzocchi M. A critical point theory for nonsmooth functionals. Ann. Matem. Pura ed Appl. 1994;IV, CLXVII:73–100.MathSciNetCrossRefGoogle Scholar
  14. [Ek1]
    Ekeland I. Nonconvex minimization problems. Bull. Amer. Math. Soc. (NS), 1979;1:443–474.MathSciNetMATHCrossRefGoogle Scholar
  15. [Ek2]
    Ekeland, Ivar. Convexiy Methods in Hamiltoniam Mechanics. N.Y.: Springer-Verlag, 1990.CrossRefGoogle Scholar
  16. [FG]
    Fang G, Ghoussoub N. Second-order information on Palais—Smale sequences in the mountain-pass theorem. Manuscr. Mathem., 1992.Google Scholar
  17. [Fig2]
    Figueredo DG. Lectures on the Ekeland variational principle with applications and detours. Preliminary Lecture Notes, SISSA, 1988.Google Scholar
  18. [FS]
    Figueredo DG, Solimini S. A variational approach to superlinear elliptic problems. Comm. Partial Differential Equations, 1984;9:699–717.MathSciNetCrossRefGoogle Scholar
  19. [Gh1]
    Ghossoub N. Location, Multiplicity and Morse indices of min-max critical points. J. Reine Angew. Math., 1991;417:27–76.MathSciNetGoogle Scholar
  20. [Gh2]
    Ghossoub N. Duality and perturbation methods in critical point theory. Cambridge University Press, 1994.Google Scholar
  21. [GP]
    Ghossoub N, Preiss D. A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henry Poincaré, 1989;6,5:321–330.Google Scholar
  22. [GSQ]
    Guo D, Sun J, Qi G. Some extensions of the mountain-pass lemma. Diff. Int. Eq., 1988;1,3:351–358.MathSciNetGoogle Scholar
  23. [Ho1]
    Hofer H. Variational and topological methods in partially ordered Hilbert spaces. Math. Ann., 1982;261:493–514.MathSciNetMATHCrossRefGoogle Scholar
  24. [Hofer H]
    The topological degree at a critical point of mountain pass type. Proc. Symp. Pure Math. I, AMS, 1986;501–509.Google Scholar
  25. [KF]
    Kolmogorov A, Fomin V. Elements of the Functional Analysis and Measure Theory. Moskow: Nauka, 1976 (in Russian).MATHGoogle Scholar
  26. [LSn]
    Lusternik L, Schnirelman L. Methodes Topologiques dans lesProblemes Variationels. Paris: Gautheir-Vilar, 1934.MATHGoogle Scholar
  27. [M1]
    Mawhin, Jean. Points Fixes, Points Critiques et Problemes aux Limites. Semin. Math. Sup. N 92, Montreal: Presses Univ.Montreal, 1985.MATHGoogle Scholar
  28. [M2]
    Mawhin J. Critical point theory and nonlinear differential equations. in Equadiff 6, Brno 1985, Lect. Notes in Math. N 1192, Springer Verlag, Berlin, 1986, 49–58.Google Scholar
  29. [MW1]
    Mawhin J, Willem M. Multiple solutions of the periodic boundary value problems for some forced pendulum type equations. J. Diff. Eq., 1984;52:264–287.MathSciNetMATHCrossRefGoogle Scholar
  30. [MW2]
    Mawhin, Jean and Willem, Michael. Critical Point Theory and Hamiltonian Systems. N.Y.: Springer-Verlag, 1988.Google Scholar
  31. [Pal]
    Palais RS. Critical point theory and the minimax principle. Proc. Sympos. Pure Math. vol.15, Amer. Math. Soc. Providence, R.I., 1970,185–212.MathSciNetCrossRefGoogle Scholar
  32. [PS1]
    Pucci P, Serrin J. Extensions of the montain-pass lemma. J. Functional Anal., 1984;59:185–210.MathSciNetMATHCrossRefGoogle Scholar
  33. [P52]
    Pucci P, Serrin J. A montain-pass lemma. J. Diff. Eq., 1985;60:142–149.MathSciNetMATHCrossRefGoogle Scholar
  34. [Ra0]
    Rabinowitz P. A note on nonlinear eigenvalue problems for a class of differential equations. J. Diff. Eq., 1971;9:536–548.MathSciNetMATHCrossRefGoogle Scholar
  35. [Ra1]
    Rabinowitz P. The mountain-pass theorem: Theme and variations. Lecture Notes in Math., N 957, N.Y.: Springer-Verlag, 1982;237–269.Google Scholar
  36. [Ra2]
    Rabinowitz P. Minimax methods in Critical Point Theory and Applications to Differential Equations. CBMS Reg. Conf. 65, AMS, Providence, R.I., 1986.Google Scholar
  37. [Ram]
    Ramos, Miguel. Teoremas de Enlace na Teoria dos Pontos Críticos. Universidade de Lisboa, Faculdade de Ciências, 1993.MATHGoogle Scholar
  38. [RTK1]
    Ribarska N., Tsachev T. and Krstanov M., The intrisic mountain pass principle. C. R. Acad. Sci. Paris, t. 329, Ser. I, 1999; 399–404.CrossRefGoogle Scholar
  39. [5a2]
    Sanchez L. Métodos da Teoria de Pontos Criticos. Universidade de Lisboa, Faculdade de Ciências, 1993.Google Scholar
  40. [Sch1]
    Schechter M. A bounded Mountain Pass Lemma without the (PS) condition. Trans. AMS, 1992;331:681–703.MathSciNetMATHCrossRefGoogle Scholar
  41. [Sch2]
    Schechter M. A variation of the Mountain Pass Theorem and applications. J. London Math. Soc., 1991;44:491–502.MathSciNetMATHCrossRefGoogle Scholar
  42. [St1]
    Struwe M. Multiple solutions of differential equations without the Palais-Smale condition. Math. Anal., 1982;261:399–412.MathSciNetMATHCrossRefGoogle Scholar
  43. [St2]
    Struwe M. Generalized Palais-Smale condition and applications, Universität Bonn, preprint N 17, 1983.Google Scholar
  44. [St3]
    Struwe, Michael. Variational Methods. N.Y.: Springer Verlag, 1990.MATHCrossRefGoogle Scholar
  45. [V]
    Vainberg, Morduchai. Variational Methods for the Study of Nonlinear Operators. San Francisco: Holden-Day, 1984.Google Scholar
  46. [Will]
    Willem M. Lecture notes on critical point theory, Fundaçâo Universidade de Brasília, 199, 1983.Google Scholar
  47. [Wi13]
    Willem, Michael. Minimax Theorems. Basel: Birkhäser, 1997.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Maria do Rosário Grossinho
    • 1
    • 2
  • Stepan Agop Tersian
    • 3
  1. 1.ISEGUniversidade Técnica de LisboaPortugal
  2. 2.CMAFUniversidade de LisboaPortugal
  3. 3.University of RousseBulgaria

Personalised recommendations