A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type

Frank Arthur; Xiaodong Yan

doi:10.3934/cpaa.2016.15.807

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2016, Volume 15, Issue 3: 807-830. Doi: 10.3934/cpaa.2016.15.807

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A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type

Frank Arthur^1, and
Xiaodong Yan^1,

1.
Department of Mathematics, University of Connecticut, Storrs, CT 06269

Received: March 31, 2015

Revised: September 30, 2015

Published: April 2016

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Abstract

We prove there are no positive solutions with slow decay rates to higher order elliptic system \begin{eqnarray} \left\{ \begin{array}{c} \left( -\Delta \right) ^{m}u=\left\vert x\right\vert ^{a}v^{p} \\ \left( -\Delta \right) ^{m}v=\left\vert x\right\vert ^{b}u^{q} \end{array} \text{ in }\mathbb{R}^{N}\right. \end{eqnarray} if $p\geq 1,$ $q\geq 1,$ $\left( p,q\right) \neq \left( 1,1\right) $ satisfies $\frac{1+\frac{a}{N}}{p+1}+\frac{1+\frac{b}{N}}{q+1}>1-\frac{2m}{N} $ and \begin{eqnarray} \max \left( \frac{2m\left( p+1\right) +a+bp}{pq-1},\frac{2m\left( q+1\right) +aq+b}{pq-1}\right) >N-2m-1. \end{eqnarray} Moreover, if $N=2m+1$ or $N=2m+2,$ this system admits no positive solutions with slow decay rates if $p\geq 1,$ $q\geq 1,$ $\left( p,q\right) \neq \left( 1,1\right) $ satisfies $\frac{1}{ p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$

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Mathematics Subject Classification: Primary: 35B53, 35J48; Secondary: 35B09, 35J61.

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