On the normalized eigenvalue problems for nonlinear elliptic operators II

This paper continues our previous research on the following form of normalized eigenvalue problemAu−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∗X∗ locally uniformly convex, D⊂XD⊂X is a bounded open set, the operator C   is defined only on R¯+×∂D such that the closure of a subset of {C(λ,u)/‖C(λ,u)‖}{C(λ,u)/‖C(λ,u)‖} is not equal to the unit sphere of X∗X∗. This research reveals the fact that such eigenvalue problems do not depend on the properties of C   located in R¯+×D. Similar result holds for the bounded, demicontinuous (S)+(S)+ operator A. This remarkable discovery is applied to the nonlinear elliptic operators under degenerate and singular conditions.