A Note on a Non-linear Krein-rutman Theorem

In this note we will present an extension of the Krein-Rutman theorem for an abstract non-linear, compact, positively 1-homogeneous operators on a Banach space having the properties of being increasing with respect to a convex cone K and such that there is a non-zero u ∈ K for which M Tu < u for some positive constant M . This will provide a uniform framework for recovering the Krein-Rutman-like theorems proved for many non-linear differential operators of elliptic type, like the pLaplacian cf. Anane [1], Hardy-Sobolev operator cf. Sreenadh [13], Pucci’s operator cf. Felmer et. al. [6]. Our proof follows the same lines as in the linear case cf. Rabinowitz [12] and is based on a bifurcation theorem.