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Article Contents

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

• 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 $\frac{1}{ p+1}+\frac{1}{q+1}>1-\frac{2m}{N}.$
Mathematics Subject Classification: Primary: 35B53, 35J48; Secondary: 35B09, 35J61.

 Citation:

•  [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.