Minimization Techniques: Compact Problems

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


Throughout this chapter we show how techniques based on minimization arguments can be used to establish existence results for various types of problems.

Our aim is not to describe the most general results, but to give a series of examples, and to show how simple techniques can be refined to treat more complex cases.

We start from sublinear problems, for which the energy functionals are bounded from below. Direct minimization arguments, based on convexity and coercivity allow one to establish rather easily the existence of a solution.

Next we turn to superlinear problems. For these types of nonlinearities direct minimization does not work anymore: the corresponding energy functionals are generally unbounded from below. We then present some methods of constrained minimization, where one restricts the functional to a subset of functions on which it is bounded from below, and tries to establish the existence of a minimum point. Special care must then be employed to show that the constrained minimum is truly a critical point of the unconstrained functional.


Weak Solution Nontrivial Solution Trivial Solution Nonlinear Eigenvalue Problem Bibliographical Note 
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. 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
  2. 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
  3. 14.
    H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983) MATHCrossRefMathSciNetGoogle Scholar
  4. 17.
    K. Deimling, Nonlinear Functional Analysis (Springer, Berlin, 1985) MATHGoogle Scholar
  5. 19.
    I. Ekeland, R. Temam, Analyse convexe et problèmes variationnels. Collection Etudes Mathématiques (Dunod/Gauthier-Villars, Paris, 1974) MATHGoogle Scholar
  6. 20.
    J.P. Garcia Azorero, I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues. Commun. Partial Differ. Equ. 12(12), 1389–1430 (1987) MATHCrossRefGoogle Scholar
  7. 21.
    E. Giusti, Direct Methods in the Calculus of Variations (World Scientific, River Edge, 2003) MATHCrossRefGoogle Scholar
  8. 26.
    O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques. Mathématiques & Applications, vol. 13 (Springer, Paris, 1993) MATHGoogle Scholar
  9. 27.
    I. Kuzin, S. Pohozaev, Entire Solutions of Semilinear Elliptic Equations. Progress in Nonlinear Differential Equations and Their Applications, vol. 33 (Birkhäuser, Basel, 1997) MATHGoogle Scholar
  10. 30.
    P. Lindqvist, Notes on the p-Laplace equation, Report, University of Jyväskylä, Department of Mathematics and Statistics, 102 (2006) Google Scholar
  11. 36.
    Z. Nehari, Characteristic values associated with a class of non-linear second-order differential equations. Acta Math. 105, 141–175 (1961) MATHCrossRefMathSciNetGoogle Scholar
  12. 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
  13. 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
  14. 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