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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

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, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317
References:
[1]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51.  Google Scholar

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D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21(1994), 387-397.  Google Scholar

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C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar

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J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.  Google Scholar

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E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbbR^n$., Diff. Int. Eq., 9 (1996), 465-479. Google Scholar

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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), 411-619. doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

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J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68.  Google Scholar

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J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653.  Google Scholar

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J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

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P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. in Math., 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

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M. A. Souto, Sobre a Existência de Soluç ões Positivas Para Sistemas Cooperativos Não Lineares, PhD thesis, Unicamp, 1992. Google Scholar

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J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

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X. Yan, A Liouville Theorem for Higher order Elliptic system, J. Math. Anal. Appl., 387 (2012), 153-165. doi: 10.1016/j.jmaa.2011.08.081.  Google Scholar

show all references

References:
[1]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51.  Google Scholar

[2]

D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21(1994), 387-397.  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), 206-231. 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), 685-709. 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), 125-151. 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), 465-479. 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), 411-619. 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), 55-68.  Google Scholar

[9]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653.  Google Scholar

[10]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[11]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. in Math., 221 (2009), 1409-1427. 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, Unicamp, 1992. Google Scholar

[13]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

[14]

X. Yan, A Liouville Theorem for Higher order Elliptic system, J. Math. Anal. Appl., 387 (2012), 153-165. doi: 10.1016/j.jmaa.2011.08.081.  Google Scholar

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