Nonlinear Elliptic Eigenvalue Problems

Part of the Texts and Monographs in Physics book series (TMP)


As a further application of the direct methods of the calculus of variations let us discuss a special class of nonlinear eigenvalue problems. As far as the technical framework is concerned, we proceed here as in our treatment of nonlinear boundary value problems in Chap. 7, which means, that if we are seeking solutions in a region G ⊂ ℝ n , we work in an appropriate Sobolev space W m,p (G) = E. The starting point of this method of solution is the following simple application of Theorem 4.2.3 on Lagrange multipliers. If f and h are two C l functions on E with the derivatives Df = f′ and Dh = h′, then we can solve the nonlinear eigenvalue equation
$$\begin{array}{*{20}{c}} {f\prime (u) = \lambda h\prime (u),} & {u \in E,} & {\lambda \in \mathbb{R},} \\ \end{array}$$
in a simple way by determining the critical points of the function h on suitable level surfaces f −1 (c) of f or, conversely, by determining the critical points of f on sutiable level surfaces h −1 (c) of h. The eigenvalue λ appears thereby as a Lagrange multiplier.


Eigenvalue Problem Level Surface Compact Manifold Real Banach Space Separable Banach Space 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 8.1
    Browder, F. E.: Nonlinear eigenvalue problems and group invariance. In: Functional analysis and related fields, F. E. Browder (ed.). Springer, Berlin Heidelberg 1970Google Scholar
  2. 8.2
    Browder, F. E.: Existence theorems for nonlinear partial differential equations. Global Analysis, Proc. Symp. Pure Math. 16 (1970) 1–62Google Scholar
  3. 8.3
    Reed, M., Simon, B.: Methods of modern mathematical physics IV. Academic Press, New York 1975Google Scholar
  4. 8.4
    Ljusternik, L.: Sur quelques méthodes topologiques en géometrie différentielle. Atti dei Congresso Internationale dei Matematici Bologna 4 (1928) 291–296Google Scholar
  5. 8.5
    Ljusternik, L., Schnirelman, T.: Méthodes topologiques dans les problèmes variationels. Hermann, Paris 1934Google Scholar
  6. 8.6
    Ljusternik, L.A., Schnirelman, L. G.: Topological methods in variational problems. Trudy Inst. Math. Mech., Moscow State University 1936, pp. 1–68Google Scholar
  7. 8.7
    Palais, R. S.: Ljusternik-Schnirelman theory on Banach manifolds. Topology 5 (1967) 115–132CrossRefMathSciNetGoogle Scholar
  8. 8.8
    Rabinowitz, P. H.: Variational methods for nonlinear elliptic eigenvalue problems. Indiana Univ. Math. J. 23 (1974) 729–754zbMATHMathSciNetGoogle Scholar
  9. 8.9
    Browder, F. E.: Non linear eigenvalue problems and Galerkin-approximation. Bull. Amer. Math. Soc. 74 (1968) 651–656CrossRefzbMATHMathSciNetGoogle Scholar
  10. 8.10
    Bröcker, T., Jänisch, K.: Einführung in die Differentialtopologie. Heidelberger Taschenbücher 143. Springer, Berlin Heidelberg 1973Google Scholar
  11. 8.11
    Lang, S.: Differential manifolds. Addison-Wesley, Reading, MA 1972zbMATHGoogle Scholar
  12. 8.12
    Palais, R. S., Smale, S.: A generalized Morse theory. Bull. Amer. Math. Soc. 70 (1964) 165–171CrossRefzbMATHMathSciNetGoogle Scholar
  13. 8.13
    Krasnoselskij, M.: Topological methods in the theory of nonlinear integral equations. Pergamon, New York 1964Google Scholar
  14. 8.14
    Coffman, C. V.: A minimum-maximum principle for a class of nonlinear integral equations. Analyse Mathématique 22 (1969) 391–419CrossRefzbMATHMathSciNetGoogle Scholar
  15. 8.15
    Dieudonné, J.: Foundations of modern analysis. Academic Press, New York 1969zbMATHGoogle Scholar
  16. 8.16
    Robertson, A.P., Robertson, W. J.: Topological vector spaces. Cambridge University Press, Cambridge 1973zbMATHGoogle Scholar
  17. 8.17
    Schwartz, J. T.: Nonlinear functional analysis. Gordon and Breach, New York 1969zbMATHGoogle Scholar
  18. 8.18
    Rabinowitz, P. H.: Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973) 161–202zbMATHMathSciNetGoogle Scholar
  19. 8.19
    Vainberg, M. M.: Variational methods for the study of nonlinear operators. Holden Day, London 1964zbMATHGoogle Scholar
  20. 8.20
    Berger, M. S.: Non linearity and functional analysis. Academic Press, New York 1977Google Scholar

Further Reading

  1. Palais, R. S.: Critical point theory and the minimax principle. Proc. Am. Math. Soc. Summer Institute on Global Analysis, S. S. Chen and S. Smale (eds.), 1968Google Scholar
  2. Rabinowitz, P. H.: Pairs of positive solutions for nonlinear elliptic partial differential equations. Indiana Univ. Math. J. 23 (1974) 173–186MathSciNetGoogle Scholar
  3. Alber, S.I.: The topology of functional manifolds and the calculus of variations in the large. Russian Mathematical Surveys 25 (1970) 4MathSciNetGoogle Scholar
  4. Amann, H.: Ljusternik-Schnirelman theory and nonlinear eigenvalue problems. Math. Ann. 199 (1972) 55–72CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

Personalised recommendations