A Liouville-type theorem for higher order elliptic systems

Previous Article
Density of fiberwise orbits in minimal iterated function systems on the circle
DCDS Home
This Issue
Next Article

September 2014, 34(9): 3317-3339. doi: 10.3934/dcds.2014.34.3317

A Liouville-type theorem for higher order elliptic systems

Frank Arthur ^1, , Xiaodong Yan ^1, and Mingfeng Zhao ^1,

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

Abstract
Related Papers

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

Keywords: Higher order elliptic systems, Liouville theorem..

Mathematics Subject Classification: Primary: 35J30, 35B53; Secondary: 35J9.

Citation: Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317

References:

[1]	J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51.
[2]	D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21(1994), 387-397.
[3]	C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{N}$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.
[4]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb{R}^{N}$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.
[5]	E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923.
[6]	E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$., Diff. Int. Eq., 9 (1996), 465-479.
[7]	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.
[8]	J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68.
[9]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653.
[10]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369-380.
[11]	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.
[12]	M. A. Souto, Sobre a Existência de Soluç ões Positivas Para Sistemas Cooperativos Não Lineares, PhD thesis, Unicamp, 1992.
[13]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.
[14]	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.

show all references

References:

[1]	J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden systems, Indiana. J., 51 (2002), 37-51.
[2]	D. G. de Figueiredo and P. Felmer, A Liouville-type Theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21(1994), 387-397.
[3]	C.-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{N}$, Comment. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.
[4]	J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb{R}^{N}$, J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.
[5]	E. Mitidieri, A Rellich type identity and applications, Comm. P.D.E., 18 (1993), 125-151. doi: 10.1080/03605309308820923.
[6]	E. Mitidieri, Non-existence of positive solutions of semilinear elliptic systems in $\mathbb{R}^{N}$., Diff. Int. Eq., 9 (1996), 465-479.
[7]	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.
[8]	J. Serrin and H. Zou, Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, 3 (1994), 55-68.
[9]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Int. Eq., 9 (1996), 635-653.
[10]	J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden systems, Atti. Sem. mat. Fis. Univ. Modena, 46 (1998), 369-380.
[11]	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.
[12]	M. A. Souto, Sobre a Existência de Soluç ões Positivas Para Sistemas Cooperativos Não Lineares, PhD thesis, Unicamp, 1992.
[13]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.
[14]	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.

[1]	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
[2]	Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361
[3]	Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134
[4]	Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[5]	Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[6]	Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
[7]	Yuan Li. Extremal solution and Liouville theorem for anisotropic elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4063-4082. doi: 10.3934/cpaa.2021144
[8]	Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035
[9]	Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
[10]	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
[11]	Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure and Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565
[12]	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
[13]	Xinmin Yang, Jin Yang, Heung Wing Joseph Lee. Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs. Journal of Industrial and Management Optimization, 2013, 9 (3) : 525-530. doi: 10.3934/jimo.2013.9.525
[14]	Yuxia Guo, Ting Liu. Liouville-type theorem for high order degenerate Lane-Emden system. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2073-2100. doi: 10.3934/dcds.2021184
[15]	Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581
[16]	Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074
[17]	Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081
[18]	Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[19]	G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89
[20]	Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Frank Arthur Xiaodong Yan Mingfeng Zhao
on Google Scholar
Frank Arthur Xiaodong Yan Mingfeng Zhao

[Back to Top]