American Institute of Mathematical Sciences

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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

Zhenjie Li 1, , Ze Cheng 2, and  Dongsheng Li 1,

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.
Keywords: weighted Hardy-Littlewood-Sobolev inequality., Louville type theorem, integral systems, method of moving planes of integral form.
Mathematics Subject Classification: Primary: 35A05, 35J45, 35J60; Secondary: 45G1.
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

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