Positive solutions of asymptotically linear elliptic eigenvalue problems

The existence of positive solutions of elliptic eigenvalue problems of the form (0.1) has been investigated extensively in recent years; we refer to the survey article of Amann [I] and the references to work by Keener and Keller, Laetsch, Dancer, and others therein. In these papers it is assumed thatfmaps Rf into R+ and, in general, the main tool in the proofs is the theory of positive operators in ordered Banach spaces. Clearly, this theory is no longer available if f is allowed to assume negative values also. We will be interested mainly in this case