On unique continuation principles for some elliptic systems

TL;DR

This paper establishes unique continuation principles for certain elliptic systems under superlinearity conditions and applies these results to prove nonexistence of nontrivial radial solutions in critical regimes, extending to nonlinear operators.

Contribution

It introduces new unique continuation principles for elliptic systems and applies them to nonlinear PDEs, including fully nonlinear operators like Pucci's, with novel nonexistence results.

Findings

Unique continuation principles for elliptic systems established.

Nonexistence of nontrivial radial solutions in critical regimes proved.

Results extend to fully nonlinear operators like Pucci's extremal operators.

Abstract

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations.