A Liouville-type theorem for higher order elliptic systems

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September 2014, 34(9): 3317-3339. doi: 10.3934/dcds.2014.34.3317

A Liouville-type theorem for higher order elliptic systems

Frank Arthur ^1, , Xiaodong Yan ^1, and Mingfeng Zhao ^1,

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

Received August 2013 Revised December 2013 Published March 2014

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

Keywords: Higher order elliptic systems, Liouville theorem..

Mathematics Subject Classification: Primary: 35J30, 35B53; Secondary: 35J9.

Citation: Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

References:

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References:

[1]	J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems,, Indiana. J., 51 (2002), 37. Google Scholar
[2]	D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems,, Ann. Sc. Norm. Sup. Pisa, 21 (1994), 387. Google Scholar
[3]	C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$,, Comment. Math. Helv., 73 (1998), 206. doi: 10.1007/s000140050052. Google Scholar
[4]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$,, J. Differential Equations, 225 (2006), 685. doi: 10.1016/j.jde.2005.10.016. Google Scholar
[5]	E. Mitidieri, A Rellich type identity and applications,, Comm. P.D.E., 18 (1993), 125. doi: 10.1080/03605309308820923. Google Scholar
[6]	E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbbR^n$.,, Diff. Int. Eq., 9 (1996), 465. Google Scholar
[7]	P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic systems,, Duke Math. J., 139 (2007), 411. doi: 10.1215/S0012-7094-07-13935-8. Google Scholar
[8]	J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems,, Discourses in Mathematics and its Applications, 3 (1994), 55. Google Scholar
[9]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, Diff. Int. Eq., 9 (1996), 635. Google Scholar
[10]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems,, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369. Google Scholar
[11]	P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, Adv. in Math., 221 (2009), 1409. doi: 10.1016/j.aim.2009.02.014. Google Scholar
[12]	M. A. Souto, Sobre a Existência de Soluç ões Positivas Para Sistemas Cooperativos Não Lineares,, PhD thesis, (1992). Google Scholar
[13]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258. Google Scholar
[14]	X. Yan, A Liouville Theorem for Higher order Elliptic system,, J. Math. Anal. Appl., 387 (2012), 153. doi: 10.1016/j.jmaa.2011.08.081. Google Scholar

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