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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 & 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,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2513.  doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

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

[3]

C. Cowan, A Liouville theorem for a fourth order Hé non equation,, \emph{Adv. Nonlinear Stud.}, 14 (2014), 767.   Google Scholar

[4]

M. Fazly, Liouville theorems for the polyharmonic Hé non-Lane-Emden system,, \emph{Methods. Appl. Anal.}, 21 (2014), 265.  doi: 10.4310/MAA.2014.v21.n2.a5.  Google Scholar

[5]

M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2513.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

E. Mitidieri, A Rellich type identity and applications,, \emph{Comm. P.D.E.}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[10]

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

[11]

Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations,, \emph{Adv. Differential Equations}, 17 (2012), 605.   Google Scholar

[12]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equaions,, \emph{J. Diff. Equ.}, 252 (2012), 2544.  doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[13]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

M. A. Souto, Sobre a existência de soluçōes positivas para sistemas cooperativos nāo lineares,, PhD thesis, (1992).   Google Scholar

[19]

J. Villavert, Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality,, \emph{J. Differential Equations}, 258 (2015), 1685.  doi: 10.1016/j.jde.2014.11.011.  Google Scholar

[20]

X. Yan, A Liouville theorem for higher order elliptic system,, \emph{J. Math. Anal. Appl.}, 387 (2012), 153.  doi: 10.1016/j.jmaa.2011.08.081.  Google Scholar

show all references

References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville theorem for higher order elliptic system,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2513.  doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

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

[3]

C. Cowan, A Liouville theorem for a fourth order Hé non equation,, \emph{Adv. Nonlinear Stud.}, 14 (2014), 767.   Google Scholar

[4]

M. Fazly, Liouville theorems for the polyharmonic Hé non-Lane-Emden system,, \emph{Methods. Appl. Anal.}, 21 (2014), 265.  doi: 10.4310/MAA.2014.v21.n2.a5.  Google Scholar

[5]

M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2513.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

E. Mitidieri, A Rellich type identity and applications,, \emph{Comm. P.D.E.}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[10]

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

[11]

Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy-Hénon equations,, \emph{Adv. Differential Equations}, 17 (2012), 605.   Google Scholar

[12]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equaions,, \emph{J. Diff. Equ.}, 252 (2012), 2544.  doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[13]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part I: Elliptic systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

M. A. Souto, Sobre a existência de soluçōes positivas para sistemas cooperativos nāo lineares,, PhD thesis, (1992).   Google Scholar

[19]

J. Villavert, Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality,, \emph{J. Differential Equations}, 258 (2015), 1685.  doi: 10.1016/j.jde.2014.11.011.  Google Scholar

[20]

X. Yan, A Liouville theorem for higher order elliptic system,, \emph{J. Math. Anal. Appl.}, 387 (2012), 153.  doi: 10.1016/j.jmaa.2011.08.081.  Google Scholar

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