Qualitative properties for solutions to conformally invariant fourth order critical systems

TL;DR

This paper investigates the qualitative behavior of nonnegative solutions to a conformally invariant fourth order system with critical exponents, classifying solutions based on singularity type and establishing symmetry and non-existence results.

Contribution

It extends classical classification results to a coupled fourth order system, identifying symmetry properties and ruling out semi-singular solutions.

Findings

Non-singular solutions are rotationally invariant and weakly positive.

Solutions with non-removable singularity are radially symmetric and strongly positive.

Semi-singular solutions do not exist; all components blow up similarly near the origin.

Abstract

We study qualitative properties for nonnegative solutions to a conformally invariant coupled system of fourth order equations involving critical exponents. For solutions defined in the punctured space, there exist essentially two cases to analyze. If the origin is a removable singularity, we prove that non-singular solutions are rotationally invariant and weakly positive. More precisely, they are the product of a fourth order spherical solution by a unit vector with nonnegative coordinates. If the origin is a non-removable singularity, we show that the solutions are radially symmetric and strongly positive. Furthermore, using a Pohozaev-type invariant, we prove the non-existence of semi-singular solutions, that is, all components equally blow-up in the neighborhood of origin. Namely, they are classified as multiples of the Emden--Fowler solution. Our results are natural generalizations of the famous classification due to [L. A. Caffarelli, B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1989)] on the classical singular Yamabe equation.