On the normalized eigenvalue problems for nonlinear elliptic operators II

This paper continues our previous research on the following form of normalized eigenvalue problem Au−C(λ,u) = 0, λ 0 and u ∈ ∂D, where the operator A is maximal monotone on an infinitely dimensional, real reflexive Banach space X with both X and its dual space X∗ locally uniformly convex, D ⊂ X is a bounded open set, the operator C is defined only on + ×∂D such that the closure of a subset of {C(λ,u)/ C(λ,u) } is not equal to the unit sphere of X∗. This research reveals the fact that such eigenvalue problems do not depend on the properties of C located in + × D. Similar result holds for the bounded, demicontinuous (S)+ operator A. This remarkable discovery is applied to the nonlinear elliptic operators under degenerate and singular conditions. © 2007 Elsevier Inc. All rights reserved.