Asymptotics for positive singular solutions to subcritical sixth order equations

TL;DR

This paper classifies the local asymptotic behavior of positive singular solutions to subcritical sixth order equations, establishing radial symmetry and analyzing their growth using advanced change of variables and monotonicity formulas.

Contribution

It introduces a new change of variables and a nonautonomous Pohozaev functional to analyze singular solutions of sixth order equations, extending previous methods.

Findings

Solutions are asymptotically radially symmetric.

A new change of variables is developed for the lower critical regime.

A nonautonomous Pohozaev functional is constructed and shown to be asymptotically monotone.

Abstract

We classify the local asymptotic behavior of positive singular solutions to a class of subcritical sixth order equations on the punctured ball. Initially, using a version of the integral moving spheres technique, we prove that solutions are asymptotically radially symmetric solutions with respect to the origin. We divide our approach into some cases concerning the growth of nonlinearity. In general, we use an Emden--Fowler change of variables to translate our problem to a cylinder. In the lower critical regime, this is not enough, thus, we need to introduce a new notion of change of variables. The difficulty is that the cylindrical PDE in this coordinate system is nonautonomous. Nonetheless, we define an associated nonautonomous Pohozaev functional, which can be proved to be asymptotically monotone. In addition, we show a priori estimates for these two functionals, from which we extract compactness properties. With this ingredients, we can perform an asymptotic analysis technique to prove our main result.