Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical exponent

TL;DR

This paper investigates the asymptotic behavior of positive singular solutions to strongly coupled critical elliptic systems near isolated singularities, extending classical results to higher dimensions and non-conformally flat metrics.

Contribution

It generalizes known asymptotic results for the Yamabe equation to coupled elliptic systems in dimensions up to five, including non-conformally flat metrics.

Findings

Singular solutions are asymptotic to Fowler type solutions in dimensions ≤ 5

Extension of classical Yamabe asymptotic properties to coupled systems

Generalization to non-conformally flat metrics

Abstract

We studied the asymptotic behavior of local solutions for strongly coupled critical elliptic systems near an isolated singularity. For the dimension less than or equal to five we prove that any singular solution is asymptotic to a rotationally symmetric Fowler type solution. This result generalizes the celebrated work due to Caffarelli, Gidas, and Spruck [1] who studied asymptotic proprieties to the classic Yamabe equation. In addition, we generalize similar results by Marques [11] for inhomogeneous context, that is, when the metric is not necessarily conformally flat.