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

A Liouville-type theorem for higher order elliptic systems

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 $\frac{1}{p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$
Citation: Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete and 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.

[2]

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

[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.

[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.

[5]

E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923.

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

show all references

References:
[1]

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

[2]

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

[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.

[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.

[5]

E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923.

[6]

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

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

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