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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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) 55–72CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Philippe Blanchard
    • 1
  • Erwin Brüning
    • 2
  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

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