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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res. Lett, 14 (2007), 373-383. doi: 10.2307/2152750.
[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.
[20]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Inve. Math, 123 (1996), 221-231. doi: 10.2307/2152750.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math Res. Lett, 14 (2007), 373-383. doi: 10.2307/2152750.
[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.
[20]	C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Inve. Math, 123 (1996), 221-231. doi: 10.2307/2152750.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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