# 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,, \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
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