March  2015, 14(2): 565-576. doi: 10.3934/cpaa.2015.14.565

The Liouville type theorem and local regularity results for nonlinear differential and integral systems

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China, China

2. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado

Received  April 2014 Revised  November 2014 Published  December 2014

In this paper we establish a Liouville type theorem for positive solutions of a class of system of integral equations. Firstly, we show the local regularity lifting result with the help of the Hardy-Littlewood-Sobolev inequality. Then by the method of moving planes in integral forms, we obtain a Liouville type theorem for this system.
Citation: 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
References:
[1]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system,, \emph{Indi. Unve. Math. J}, 51 (2002), 37. doi: 10.2307/2152750. Google Scholar

[2]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems,, \emph{Milan. J. Math.}, 76 (2008), 27. doi: 10.2307/2152750. Google Scholar

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure. Appl. Math.}, 42 (1989), 271. doi: 10.2307/2152750. Google Scholar

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J}, 63 (1991), 615. doi: 10.2307/2152750. Google Scholar

[5]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, \emph{Acta. Math. Scie}, 29B (2009), 949. doi: 10.2307/2152750. Google Scholar

[6]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, \emph{Comm. Pure. Appl. Anna}, 12 (2013), 2497. doi: 10.2307/2152750. Google Scholar

[7]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Disc. Cont. Dyna. Syst-A}, 24 (2009), 1167. doi: 10.2307/2152750. Google Scholar

[8]

W Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure. Appl. Math}, 59 (2006), 330. doi: 10.2307/2152750. Google Scholar

[9]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. Part. Diff. Equa}, 30 (2005), 59. doi: 10.2307/2152750. Google Scholar

[10]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Disc. Cont. Dyna. Syst-A}, 12 (2005), 347. doi: 10.2307/2152750. Google Scholar

[11]

A. Chang and P. Yang, On uniqueness of an $n$-th order differential equation in conformal geometry,, \emph{Math. Res. Lett}, 4 (1997), 91. doi: 10.2307/2152750. Google Scholar

[12]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, \emph{Rev Mat Iber}, 20 (2004), 67. doi: 10.2307/2152750. Google Scholar

[13]

P. H. Fowler, Further studies of emden's and similar differential equations,, \emph{Quar. J. Math (Oxford)}, 2 (1931), 259. doi: 10.2307/2152750. Google Scholar

[14]

D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems,, \emph{Anna. Scuola Norm. Sup. Pisa}, 21 (1994), 387. doi: 10.2307/2152750. Google Scholar

[15]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$,, \emph{Comm. Part. Diff. Equa}, 33 (2008), 263. doi: 10.2307/2152750. Google Scholar

[16]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbbR^n$,, collected in the book Mathematical Analysis and Applications, (1981). doi: 10.1007/978-1-4612-0873-0. Google Scholar

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure. Appl. Math}, 34 (1981), 525. doi: 10.2307/2152750. Google Scholar

[18]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, \emph{Math Res. Lett}, 14 (2007), 373. doi: 10.2307/2152750. Google Scholar

[19]

T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett}, 86 (2001), 5043. doi: 10.2307/2152750. Google Scholar

[20]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Inve. Math}, 123 (1996), 221. doi: 10.2307/2152750. Google Scholar

[21]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$,, \emph{Comm. Math. Helv}, 73 (1998), 206. doi: 10.2307/2152750. Google Scholar

[22]

J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$,, \emph{J. Diff. Equa}, 225 (2006), 685. doi: 10.2307/2152750. Google Scholar

[23]

C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, \emph{SIAM J. Math. Anal}, 40 (2008), 1049. doi: 10.2307/2152750. Google Scholar

[24]

T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n\leq3$,, \emph{Comm. Math. Phys}, 255 (2005), 629. doi: 10.2307/2152750. Google Scholar

[25]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, \emph{Ann Inst H Poincar\'e Anal Non-Lin}, 22 (2005), 403. doi: 10.2307/2152750. Google Scholar

[26]

E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbbR^n$,, \emph{Diff. Inte. Equa}, 9 (1996), 465. doi: 10.2307/2152750. Google Scholar

[27]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Advances in Math}, 226 (2011), 2676. doi: 10.2307/2152750. Google Scholar

[28]

L. Ma and L. Zhao, Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system,, \emph{J. Math. phys}, 49 (2008). doi: 10.2307/2152750. Google Scholar

[29]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear prooblems via Liouville-type theorems. Part I: elliptic systems,, \emph{Duke Math. J}, 139 (2007), 555. doi: 10.2307/2152750. Google Scholar

[30]

W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres,, \emph{J. Diff. Equa}, 161 (2000), 219. doi: 10.2307/2152750. Google Scholar

[31]

Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Math}, 221 (2009), 1409. doi: 10.2307/2152750. Google Scholar

[32]

E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space,, \emph{J. Math. Mech}, (1958), 503. doi: 10.2307/2152750. Google Scholar

[33]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, \emph{Diff. Inte. Equa}, 9 (1996), 635. doi: 10.2307/2152750. Google Scholar

[34]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, \emph{Atti Sem. Mat. Fis. Univ. Modena. Sippl}, 46 (1996), 369. doi: 10.2307/2152750. Google Scholar

[35]

J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system,, \emph{Comm. Part. Diff. Equa}, 23 (1998), 577. doi: 10.2307/2152750. Google Scholar

[36]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Anna}, 313 (1999), 207. doi: 10.2307/2152750. Google Scholar

[37]

X. Yu, Liouville type theorems for integral equations and integral systems,, \emph{Calc. Vari. Part. Diff. Equa}, 46 (2013), 75. doi: 10.2307/2152750. Google Scholar

show all references

References:
[1]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system,, \emph{Indi. Unve. Math. J}, 51 (2002), 37. doi: 10.2307/2152750. Google Scholar

[2]

G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems,, \emph{Milan. J. Math.}, 76 (2008), 27. doi: 10.2307/2152750. Google Scholar

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure. Appl. Math.}, 42 (1989), 271. doi: 10.2307/2152750. Google Scholar

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J}, 63 (1991), 615. doi: 10.2307/2152750. Google Scholar

[5]

W. Chen and C. Li, Classification of positive solutions for nonlinear differential and integral systems with critical exponents,, \emph{Acta. Math. Scie}, 29B (2009), 949. doi: 10.2307/2152750. Google Scholar

[6]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications,, \emph{Comm. Pure. Appl. Anna}, 12 (2013), 2497. doi: 10.2307/2152750. Google Scholar

[7]

W. Chen and C. Li, An integral system and the Lane-Emden conjecture,, \emph{Disc. Cont. Dyna. Syst-A}, 24 (2009), 1167. doi: 10.2307/2152750. Google Scholar

[8]

W Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure. Appl. Math}, 59 (2006), 330. doi: 10.2307/2152750. Google Scholar

[9]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Comm. Part. Diff. Equa}, 30 (2005), 59. doi: 10.2307/2152750. Google Scholar

[10]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, \emph{Disc. Cont. Dyna. Syst-A}, 12 (2005), 347. doi: 10.2307/2152750. Google Scholar

[11]

A. Chang and P. Yang, On uniqueness of an $n$-th order differential equation in conformal geometry,, \emph{Math. Res. Lett}, 4 (1997), 91. doi: 10.2307/2152750. Google Scholar

[12]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains,, \emph{Rev Mat Iber}, 20 (2004), 67. doi: 10.2307/2152750. Google Scholar

[13]

P. H. Fowler, Further studies of emden's and similar differential equations,, \emph{Quar. J. Math (Oxford)}, 2 (1931), 259. doi: 10.2307/2152750. Google Scholar

[14]

D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems,, \emph{Anna. Scuola Norm. Sup. Pisa}, 21 (1994), 387. doi: 10.2307/2152750. Google Scholar

[15]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$,, \emph{Comm. Part. Diff. Equa}, 33 (2008), 263. doi: 10.2307/2152750. Google Scholar

[16]

B. Gidas, W. Ni and L. Nirenberg, Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbbR^n$,, collected in the book Mathematical Analysis and Applications, (1981). doi: 10.1007/978-1-4612-0873-0. Google Scholar

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, \emph{Comm. Pure. Appl. Math}, 34 (1981), 525. doi: 10.2307/2152750. Google Scholar

[18]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, \emph{Math Res. Lett}, 14 (2007), 373. doi: 10.2307/2152750. Google Scholar

[19]

T. Kanna and M. Lakshmanan, Exact soliton solutions, shape changing collisions and partially coherent solitons in coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett}, 86 (2001), 5043. doi: 10.2307/2152750. Google Scholar

[20]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations,, \emph{Inve. Math}, 123 (1996), 221. doi: 10.2307/2152750. Google Scholar

[21]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$,, \emph{Comm. Math. Helv}, 73 (1998), 206. doi: 10.2307/2152750. Google Scholar

[22]

J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$,, \emph{J. Diff. Equa}, 225 (2006), 685. doi: 10.2307/2152750. Google Scholar

[23]

C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents,, \emph{SIAM J. Math. Anal}, 40 (2008), 1049. doi: 10.2307/2152750. Google Scholar

[24]

T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n\leq3$,, \emph{Comm. Math. Phys}, 255 (2005), 629. doi: 10.2307/2152750. Google Scholar

[25]

T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations,, \emph{Ann Inst H Poincar\'e Anal Non-Lin}, 22 (2005), 403. doi: 10.2307/2152750. Google Scholar

[26]

E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbbR^n$,, \emph{Diff. Inte. Equa}, 9 (1996), 465. doi: 10.2307/2152750. Google Scholar

[27]

C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type,, \emph{Advances in Math}, 226 (2011), 2676. doi: 10.2307/2152750. Google Scholar

[28]

L. Ma and L. Zhao, Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system,, \emph{J. Math. phys}, 49 (2008). doi: 10.2307/2152750. Google Scholar

[29]

P. Poláčik, P. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear prooblems via Liouville-type theorems. Part I: elliptic systems,, \emph{Duke Math. J}, 139 (2007), 555. doi: 10.2307/2152750. Google Scholar

[30]

W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres,, \emph{J. Diff. Equa}, 161 (2000), 219. doi: 10.2307/2152750. Google Scholar

[31]

Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Math}, 221 (2009), 1409. doi: 10.2307/2152750. Google Scholar

[32]

E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space,, \emph{J. Math. Mech}, (1958), 503. doi: 10.2307/2152750. Google Scholar

[33]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system,, \emph{Diff. Inte. Equa}, 9 (1996), 635. doi: 10.2307/2152750. Google Scholar

[34]

J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system,, \emph{Atti Sem. Mat. Fis. Univ. Modena. Sippl}, 46 (1996), 369. doi: 10.2307/2152750. Google Scholar

[35]

J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system,, \emph{Comm. Part. Diff. Equa}, 23 (1998), 577. doi: 10.2307/2152750. Google Scholar

[36]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Anna}, 313 (1999), 207. doi: 10.2307/2152750. Google Scholar

[37]

X. Yu, Liouville type theorems for integral equations and integral systems,, \emph{Calc. Vari. Part. Diff. Equa}, 46 (2013), 75. doi: 10.2307/2152750. Google Scholar

[1]

Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164

[2]

Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791

[3]

Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027

[4]

Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951

[5]

Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987

[6]

Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935

[7]

Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653

[8]

Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171

[9]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061

[10]

Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057

[11]

Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015

[12]

Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure & Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018

[13]

Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure & Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008

[14]

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

[15]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138

[16]

Xiaohui Yu. Liouville type theorems for singular integral equations and integral systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1825-1840. doi: 10.3934/cpaa.2016017

[17]

Douglas A. Leonard. A weighted module view of integral closures of affine domains of type I. Advances in Mathematics of Communications, 2009, 3 (1) : 1-11. doi: 10.3934/amc.2009.3.1

[18]

Dorota Bors, Andrzej Skowron, Stanisław Walczak. Systems described by Volterra type integral operators. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2401-2416. doi: 10.3934/dcdsb.2014.19.2401

[19]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[20]

Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]