Classification for positive singular solutions to critical sixth order equations

TL;DR

This paper classifies positive singular solutions to critical sixth order equations in punctured space, revealing they are composed of a radial singular part and a periodic solution, using advanced ODE and integral methods.

Contribution

It introduces a novel classification of solutions for sixth order equations with singularities, employing a combination of integral sliding, symmetry, and energy conservation techniques.

Findings

Solutions are radially symmetric around the singularity.

Solutions can be expressed as a product of a singular radial factor and a periodic function.

The classification includes uniqueness, boundedness, and asymptotic behavior of solutions.

Abstract

We classify entire positive singular solutions to a family of critical sixth order equations in the punctured space with a non-removable singularity at the origin. More precisely, we show that when the origin is a non-removable singularity, solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients. On the technical level, we combine integral sliding methods and qualitative analysis of ODEs, based on a conservation of energy result, to perform a topological two-parameter shooting technique. We first use the integral representation of our equation to run a moving spheres technique, which proves that solutions are radially symmetric with respect to the origin. Thus, in Emden--Fowler coordinates, we can reduce our problem to the study of an sixth order autonomous ODE with constant coefficients. The main heuristics behind our arguments is that since all the indicial roots of the ODE operator are positive, it can be decomposed into the composition of three second order operators satisfying a comparison principle. This allows us to define a Hamiltonian energy which is conserved along solutions, from which we extract their qualitative properties, such as uniqueness, boundedness, asymptotic behavior, and classification.