On the normalized eigenvalue problems for nonlinear elliptic operators

Abstract

This paper solves the following form of normalized eigenvalue problem: A u − 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 R ¯ + × ∂ 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 R ¯ + × 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.