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⩾0andu∈∂D, where DD is a bounded open set in a real infinite-dimensional Banach space X and both X   and its dual X∗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)‖}{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.