    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 & Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317
##### References:
  J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51. Google Scholar  D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21(1994), 387-397. Google Scholar  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.  Google Scholar  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.  Google Scholar  E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar  E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbbR^n$., Diff. Int. Eq., 9 (1996), 465-479. Google Scholar  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.  Google Scholar  J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68. Google Scholar  J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653. Google Scholar  J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar  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.  Google Scholar  M. A. Souto, Sobre a Existência de Soluç ões Positivas Para Sistemas Cooperativos Não Lineares, PhD thesis, Unicamp, 1992. Google Scholar  J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar  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.  Google Scholar

show all references

##### References:
  J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51. Google Scholar  D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21(1994), 387-397. Google Scholar  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.  Google Scholar  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.  Google Scholar  E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar  E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbbR^n$., Diff. Int. Eq., 9 (1996), 465-479. Google Scholar  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.  Google Scholar  J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68. Google Scholar  J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653. Google Scholar  J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369-380. Google Scholar  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.  Google Scholar  M. A. Souto, Sobre a Existência de Soluç ões Positivas Para Sistemas Cooperativos Não Lineares, PhD thesis, Unicamp, 1992. Google Scholar  J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar  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.  Google Scholar
  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  Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361  Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134  Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227  Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067  Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549  Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144  Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035  Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947  Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, 2021, 29 (5) : 2829-2839. doi: 10.3934/era.2021016  Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565  Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990  Xinmin Yang, Jin Yang, Heung Wing Joseph Lee. Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs. Journal of Industrial & Management Optimization, 2013, 9 (3) : 525-530. doi: 10.3934/jimo.2013.9.525  Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021184  Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581  Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074  Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations & Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081  Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855  G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89  Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183

2020 Impact Factor: 1.392