Principal spectral curves for Lane–Emden fully nonlinear type systems and applications

Abstract

In this paper we exploit the phenomenon of two principal half eigenvalues in the context of fully nonlinear Lane–Emden type systems with possibly unbounded coefficients and weights. We show that this gives rise to the existence of two principal spectral curves on the plane. We develop an anti-maximum principle, which is a novelty even for Lane–Emden systems involving the Laplacian operator. As applications, we derive a maximum principle in small domains for these systems, as well as existence and uniqueness of positive solutions in the sublinear regime. Most of our results are new even in the scalar case, in particular for a class of Isaac’s operators with unbounded coefficients, whose $$W^{2,\varrho }$$ regularity estimates we also prove.