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

TL;DR

This paper investigates the spectral properties of fully nonlinear Lane-Emden systems, revealing multiple principal spectral curves and establishing new maximum principles and solution existence results, even for linear cases.

Contribution

It introduces the concept of multiple principal spectral curves for nonlinear Lane-Emden systems and extends spectral theory and maximum principles to unbounded coefficient cases.

Findings

Existence of two principal spectral curves for the systems.

Construction of a third spectral curve related to a second eigenvalue.

Derivation of maximum principles and positive solution existence in sublinear regimes.

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 also construct a possible third spectral curve related to a second eigenvalue and an anti-maximum principle, which are novelties even for Lane-Emden systems involving linear operators. 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.