Classification of solutions to a system of $ n^{\rm th} $ order equations on $  \mathbb R^n $

Mathew Gluck

doi:10.3934/cpaa.2020246

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2020, Volume 19, Issue 12: 5413-5436. Doi: 10.3934/cpaa.2020246

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Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $

Mathew Gluck

Department of Mathematics, Towson University, Towson, MD 21252, USA

Received: September 30, 2019

Revised: July 31, 2020

Early access: October 2020

Published: December 2020

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Abstract

This work concerns the distributional solutions of a conformally invariant system of $ n^{\rm th} $-order elliptic equations on $ \mathbb R^n $ having exponential type nonlinearity. The system in question is a natural generalization of the constant $ Q $-curvature equation on $ \mathbb R^n $. Under an $ L^1 $-finiteness assumption and some assumptions on the coupling coefficients, an asymptotic estimate for solutions as $ \left|x\right|\to \infty $ is obtained. Under a growth constraint and further $ L^1 $-norm assumptions the method of moving spheres is used to show that, up to an additive polynomial of low degree, each of the unknown functions is a standard bubble with common center and scale parameters.

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Mathematics Subject Classification: 35C05, 35G50, 35J48.

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