Nonlinear Elliptic Eigenvalue Problems

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 ⊂ ℝⁿ, 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.