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May  2016, 15(3): 807-830. doi: 10.3934/cpaa.2016.15.807

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

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

Received  April 2015 Revised  October 2015 Published  February 2016

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 $\frac{1}{ p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$
Citation: Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure and Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
##### References:
 [1] F. Arthur, X. Yan and M. Zhao, A Liouville theorem for higher order elliptic system, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533. doi: 10.3934/dcds.2014.34.3317. [2] J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51. [3] C. Cowan, A Liouville theorem for a fourth order Hé non equation, Adv. Nonlinear Stud., 14 (2014), 767-776. [4] M. Fazly, Liouville theorems for the polyharmonic Hé non-Lane-Emden system, Methods. Appl. Anal., 21 (2014), 265-281. doi: 10.4310/MAA.2014.v21.n2.a5. [5] M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533. doi: 10.3934/dcds.2014.34.2513. [6] P. Felmer and D. G. de Figueiredo, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, XXI (1994), 387-397. [7] 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. [8] 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. [9] E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923. [10] E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbbR^n$, Diff. Int. Eq., 9 (1996), 465-479. [11] Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations, Adv. Differential Equations, 17 (2012), 605-634. [12] Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equaions, J. Diff. Equ., 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022. [13] 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), 555-579. doi: 10.1215/S0012-7094-07-13935-8. [14] J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68. [15] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653. [16] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti Sem. mat. Fis. Univ. Modena, 46 suppl (1998), 369-380. [17] 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. [18] M. A. Souto, Sobre a existência de soluçōes positivas para sistemas cooperativos nāo lineares, PhD thesis, Unicamp (1992). [19] J. Villavert, Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality, J. Differential Equations, 258 (2015), 1685-1714. doi: 10.1016/j.jde.2014.11.011. [20] 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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##### References:
 [1] F. Arthur, X. Yan and M. Zhao, A Liouville theorem for higher order elliptic system, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533. doi: 10.3934/dcds.2014.34.3317. [2] J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51. [3] C. Cowan, A Liouville theorem for a fourth order Hé non equation, Adv. Nonlinear Stud., 14 (2014), 767-776. [4] M. Fazly, Liouville theorems for the polyharmonic Hé non-Lane-Emden system, Methods. Appl. Anal., 21 (2014), 265-281. doi: 10.4310/MAA.2014.v21.n2.a5. [5] M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533. doi: 10.3934/dcds.2014.34.2513. [6] P. Felmer and D. G. de Figueiredo, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, XXI (1994), 387-397. [7] 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. [8] 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. [9] E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923. [10] E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbbR^n$, Diff. Int. Eq., 9 (1996), 465-479. [11] Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations, Adv. Differential Equations, 17 (2012), 605-634. [12] Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equaions, J. Diff. Equ., 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022. [13] 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), 555-579. doi: 10.1215/S0012-7094-07-13935-8. [14] J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68. [15] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653. [16] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti Sem. mat. Fis. Univ. Modena, 46 suppl (1998), 369-380. [17] 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. [18] M. A. Souto, Sobre a existência de soluçōes positivas para sistemas cooperativos nāo lineares, PhD thesis, Unicamp (1992). [19] J. Villavert, Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality, J. Differential Equations, 258 (2015), 1685-1714. doi: 10.1016/j.jde.2014.11.011. [20] 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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