Existence, nonexistence and uniqueness for Lane-Emden type fully nonlinear systems

TL;DR

This paper investigates the conditions for existence, nonexistence, and uniqueness of positive radial solutions in nonlinear systems driven by Pucci extremal operators, extending classical results for Lane-Emden systems.

Contribution

It introduces new regions of existence and nonexistence for solutions and establishes an exclusion principle based on trajectory analysis, including for the standard Lane-Emden system.

Findings

Identified new existence and nonexistence regions for solutions.

Proved the critical hyperbola as a threshold for solutions in Lane-Emden systems.

Discovered singular solutions with rapid decay at infinity.

Abstract

We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane-Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in $\mathbb{R}^N$ or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard Lane-Emden system involving the Laplacian operator, we prove that the critical hyperbola of regular radial positive solutions is also the threshold for existence and nonexistence of radial exterior domain solutions with Neumann boundary condition. As a byproduct, singular solutions with fast decay at infinity are also found.