# American Institute of Mathematical Sciences

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, Indi. Unve. Math. J, 51 (2002), 37-51. 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, Milan. J. Math., 76 (2008), 27-67. 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, Comm. Pure. Appl. Math., 42 (1989), 271-297. doi: 10.2307/2152750.  Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J, 63 (1991), 615-622. 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, Acta. Math. Scie, 29B (2009), 949-960. doi: 10.2307/2152750.  Google Scholar [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure. Appl. Anna, 12 (2013), 2497-2514. doi: 10.2307/2152750.  Google Scholar [7] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyna. Syst-A, 24 (2009), 1167-1184. doi: 10.2307/2152750.  Google Scholar [8] W Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math, 59 (2006), 330-343. doi: 10.2307/2152750.  Google Scholar [9] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Part. Diff. Equa, 30 (2005), 59-65. doi: 10.2307/2152750.  Google Scholar [10] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyna. Syst-A, 12 (2005), 347-354. doi: 10.2307/2152750.  Google Scholar [11] A. Chang and P. Yang, On uniqueness of an $n$-th order differential equation in conformal geometry, Math. Res. Lett, 4 (1997), 91-102. doi: 10.2307/2152750.  Google Scholar [12] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev Mat Iber, 20 (2004), 67-86. doi: 10.2307/2152750.  Google Scholar [13] P. H. Fowler, Further studies of emden's and similar differential equations, Quar. J. Math (Oxford), 2 (1931), 259-288. doi: 10.2307/2152750.  Google Scholar [14] D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa, 21 (1994), 387-397. doi: 10.2307/2152750.  Google Scholar [15] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Part. Diff. Equa, 33 (2008), 263-284. 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, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 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, Comm. Pure. Appl. Math, 34 (1981), 525-598. doi: 10.2307/2152750.  Google Scholar [18] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res. Lett, 14 (2007), 373-383. 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, Phys. Rev. Lett, 86 (2001), 5043-5046. doi: 10.2307/2152750.  Google Scholar [20] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Inve. Math, 123 (1996), 221-231. doi: 10.2307/2152750.  Google Scholar [21] C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comm. Math. Helv, 73 (1998), 206-231. doi: 10.2307/2152750.  Google Scholar [22] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$, J. Diff. Equa, 225 (2006), 685-709. doi: 10.2307/2152750.  Google Scholar [23] C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Math. Anal, 40 (2008), 1049-1057. doi: 10.2307/2152750.  Google Scholar [24] T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n\leq3$, Comm. Math. Phys, 255 (2005), 629-653. doi: 10.2307/2152750.  Google Scholar [25] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann Inst H Poincaré Anal Non-Lin, 22 (2005), 403-439. doi: 10.2307/2152750.  Google Scholar [26] E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbbR^n$, Diff. Inte. Equa, 9 (1996), 465-479. doi: 10.2307/2152750.  Google Scholar [27] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math, 226 (2011), 2676-2699. 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, J. Math. phys, 49, (2008), 062103. 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, Duke Math. J, 139 (2007), 555-579. doi: 10.2307/2152750.  Google Scholar [30] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Diff. Equa, 161 (2000), 219-243. doi: 10.2307/2152750.  Google Scholar [31] Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Math, 221 (2009), 1409-1427. doi: 10.2307/2152750.  Google Scholar [32] E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space, J. Math. Mech, 7 (1958), 503-514. doi: 10.2307/2152750.  Google Scholar [33] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Diff. Inte. Equa, 9 (1996), 635-653. doi: 10.2307/2152750.  Google Scholar [34] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl, 46 (1996), 369-380. doi: 10.2307/2152750.  Google Scholar [35] J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. Part. Diff. Equa, 23 (1998), 577-599. doi: 10.2307/2152750.  Google Scholar [36] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Anna, 313 (1999), 207-228. doi: 10.2307/2152750.  Google Scholar [37] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Vari. Part. Diff. Equa, 46 (2013), 75-95. doi: 10.2307/2152750.  Google Scholar

show all references

##### References:
 [1] J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indi. Unve. Math. J, 51 (2002), 37-51. 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, Milan. J. Math., 76 (2008), 27-67. 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, Comm. Pure. Appl. Math., 42 (1989), 271-297. doi: 10.2307/2152750.  Google Scholar [4] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J, 63 (1991), 615-622. 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, Acta. Math. Scie, 29B (2009), 949-960. doi: 10.2307/2152750.  Google Scholar [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Comm. Pure. Appl. Anna, 12 (2013), 2497-2514. doi: 10.2307/2152750.  Google Scholar [7] W. Chen and C. Li, An integral system and the Lane-Emden conjecture, Disc. Cont. Dyna. Syst-A, 24 (2009), 1167-1184. doi: 10.2307/2152750.  Google Scholar [8] W Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure. Appl. Math, 59 (2006), 330-343. doi: 10.2307/2152750.  Google Scholar [9] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Part. Diff. Equa, 30 (2005), 59-65. doi: 10.2307/2152750.  Google Scholar [10] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyna. Syst-A, 12 (2005), 347-354. doi: 10.2307/2152750.  Google Scholar [11] A. Chang and P. Yang, On uniqueness of an $n$-th order differential equation in conformal geometry, Math. Res. Lett, 4 (1997), 91-102. doi: 10.2307/2152750.  Google Scholar [12] L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev Mat Iber, 20 (2004), 67-86. doi: 10.2307/2152750.  Google Scholar [13] P. H. Fowler, Further studies of emden's and similar differential equations, Quar. J. Math (Oxford), 2 (1931), 259-288. doi: 10.2307/2152750.  Google Scholar [14] D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Anna. Scuola Norm. Sup. Pisa, 21 (1994), 387-397. doi: 10.2307/2152750.  Google Scholar [15] Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Part. Diff. Equa, 33 (2008), 263-284. 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, which is vol. 7a of the book series Advances in Mathematics. Supplementary Studies, Academic Press, New York, 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, Comm. Pure. Appl. Math, 34 (1981), 525-598. doi: 10.2307/2152750.  Google Scholar [18] F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res. Lett, 14 (2007), 373-383. 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, Phys. Rev. Lett, 86 (2001), 5043-5046. doi: 10.2307/2152750.  Google Scholar [20] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Inve. Math, 123 (1996), 221-231. doi: 10.2307/2152750.  Google Scholar [21] C. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comm. Math. Helv, 73 (1998), 206-231. doi: 10.2307/2152750.  Google Scholar [22] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbbR^n$, J. Diff. Equa, 225 (2006), 685-709. doi: 10.2307/2152750.  Google Scholar [23] C. Li and L. Ma, Uniqueness of positive bound states to Shrödinger systems with critical exponents, SIAM J. Math. Anal, 40 (2008), 1049-1057. doi: 10.2307/2152750.  Google Scholar [24] T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in $\mathbbR^n$, $n\leq3$, Comm. Math. Phys, 255 (2005), 629-653. doi: 10.2307/2152750.  Google Scholar [25] T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann Inst H Poincaré Anal Non-Lin, 22 (2005), 403-439. doi: 10.2307/2152750.  Google Scholar [26] E. Mitidieri, Non-existence of positive solutions of semilinear systems in $\mathbbR^n$, Diff. Inte. Equa, 9 (1996), 465-479. doi: 10.2307/2152750.  Google Scholar [27] C. Ma, W. Chen and C. Li, Regularity of solutions for an integral system of Wolff type, Advances in Math, 226 (2011), 2676-2699. 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, J. Math. phys, 49, (2008), 062103. 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, Duke Math. J, 139 (2007), 555-579. doi: 10.2307/2152750.  Google Scholar [30] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Diff. Equa, 161 (2000), 219-243. doi: 10.2307/2152750.  Google Scholar [31] Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Advances in Math, 221 (2009), 1409-1427. doi: 10.2307/2152750.  Google Scholar [32] E. M. Stein and G. Weiss, Fractional integrals in n-dimensional Euclidean space, J. Math. Mech, 7 (1958), 503-514. doi: 10.2307/2152750.  Google Scholar [33] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden system, Diff. Inte. Equa, 9 (1996), 635-653. doi: 10.2307/2152750.  Google Scholar [34] J. Serrin and H. Zou, Existence of positive solutions of Lane-Emden system, Atti Sem. Mat. Fis. Univ. Modena. Sippl, 46 (1996), 369-380. doi: 10.2307/2152750.  Google Scholar [35] J. Serrin and H. Zou, The existence of positive entire solutions of elliptic Hamiltonian system, Comm. Part. Diff. Equa, 23 (1998), 577-599. doi: 10.2307/2152750.  Google Scholar [36] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Anna, 313 (1999), 207-228. doi: 10.2307/2152750.  Google Scholar [37] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Vari. Part. Diff. Equa, 46 (2013), 75-95. 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] Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018 [3] Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791 [4] 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 [5] Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951 [6] 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, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987 [7] Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935 [8] 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 [9] Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171 [10] 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 [11] Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057 [12] Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022 [13] 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 [14] 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 [15] 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 [16] 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 [17] Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074 [18] José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138 [19] 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 [20] 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

2020 Impact Factor: 1.916