On the normalized eigenvalue problems for nonlinear elliptic operators

This paper solves the following form of normalized eigenvalue problem: Au−C(λ,u) = 0, λ 0 and u ∈ ∂D, where D is a bounded open set in a real infinite-dimensional Banach space X and both X and its dual X∗ are locally uniformly convex, A is an unbounded maximal monotone operator on X, the operators C is defined and continuous only on ¯ + × ∂D such that zero is not in the weak closure of a subset of {C(λ,u)/ C(λ,u) }. This research reveals the fact that such eigenvalue problems do not depend on any properties of C located in ¯ + × D. This remarkable discovery extends Theorem 4 in [A.G. Kartsatos, I.V. Skrypnik, Normalized eigenvalues for nonlinear abstract and elliptic operators, J. Differential Equations 155 (1999) 443–475] and is applied to the nonlinear elliptic operators under degenerate and singular conditions. © 2006 Elsevier Inc. All rights reserved