Nonlinear Elliptic Eigenvalue Problems

• Philippe Blanchard
• Erwin Brüning
Part of the Texts and Monographs in Physics book series (TMP)

Abstract

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
$$\begin{array}{*{20}{c}} {f\prime (u) = \lambda h\prime (u),} & {u \in E,} & {\lambda \in \mathbb{R},} \\ \end{array}$$
(8.1.1)
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.

Keywords

Manifold Assure Topo

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References

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) 115–132
8. 8.8
Rabinowitz, P. H.: Variational methods for nonlinear elliptic eigenvalue problems. Indiana Univ. Math. J. 23 (1974) 729–754
9. 8.9
Browder, F. E.: Non linear eigenvalue problems and Galerkin-approximation. Bull. Amer. Math. Soc. 74 (1968) 651–656
10. 8.10
Bröcker, T., Jänisch, K.: Einführung in die Differentialtopologie. Heidelberger Taschenbücher 143. Springer, Berlin Heidelberg 1973Google Scholar
11. 8.11
12. 8.12
Palais, R. S., Smale, S.: A generalized Morse theory. Bull. Amer. Math. Soc. 70 (1964) 165–171
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) 391–419
15. 8.15
Dieudonné, J.: Foundations of modern analysis. Academic Press, New York 1969
16. 8.16
Robertson, A.P., Robertson, W. J.: Topological vector spaces. Cambridge University Press, Cambridge 1973
17. 8.17
Schwartz, J. T.: Nonlinear functional analysis. Gordon and Breach, New York 1969
18. 8.18
Rabinowitz, P. H.: Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973) 161–202
19. 8.19
Vainberg, M. M.: Variational methods for the study of nonlinear operators. Holden Day, London 1964
20. 8.20
Berger, M. S.: Non linearity and functional analysis. Academic Press, New York 1977Google Scholar

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) 173–186
3. Alber, S.I.: The topology of functional manifolds and the calculus of variations in the large. Russian Mathematical Surveys 25 (1970) 4
4. Amann, H.: Ljusternik-Schnirelman theory and nonlinear eigenvalue problems. Math. Ann. 199 (1972) 55–72