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

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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

Frank Arthur ^1, and Xiaodong Yan ^1,

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

Received April 2015 Revised October 2015 Published February 2016

Abstract
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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 $\frac{1}{ p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$

Keywords: Liouville theorem, Hénon Lane Emden, positive solutions..

Mathematics Subject Classification: Primary: 35B53, 35J48; Secondary: 35B09, 35J6.

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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