Qualitative properties for solutions to subcritical fourth order systems

TL;DR

This paper establishes qualitative properties of singular solutions to a class of strongly coupled fourth order systems with nonlinearities similar to Gross--Pitaevskii, extending classical results and introducing new analytical tools like a nonautonomous Pohozaev functional.

Contribution

It provides a classification of blow-up solutions and asymptotic analysis for fourth order coupled systems, extending classical second order results with novel methods.

Findings

Classified limit blow-up solutions using the moving sphere method.

Proved solutions are local models near isolated singularities.

Introduced a new monotonicity-based Pohozaev functional.

Abstract

We prove some qualitative properties for singular solutions to a class of strongly coupled system involving a Gross--Pitaevskii-type nonlinearity. Our main theorems are vectorial fourth order counterparts of the classical results of [J. Serrin, Acta Math. (1964)], [P.-L. Lions, J. Differential Equations (1980)], [P. Aviles, Comm. Math. Phys. (1987)], and [B. Gidas and J. Spruck, Comm. Pure Appl. Math. (1981)]. On the technical level, we use the moving sphere method to classify the limit blow-up solutions to our system. Besides, applying asymptotic analysis techniques, we show that these solutions are indeed the local models near the isolated singularity. We also introduce a new fourth order nonautonomous Pohozaev functional, whose monotonicity properties yield improvement for the asymptotics results due to [R. Soranzo, Potential Anal. (1997)].