American Institute of Mathematical Sciences

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March  2014, 13(2): 511-525. doi: 10.3934/cpaa.2014.13.511

Liouville type theorem to an integral system in the half-space

Weiwei Zhao 1, , Jinge Yang 1, and  Sining Zheng 2,

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China, China
2. 

Department of Applied Mathematics, Dalian University of Technology, Dalian 116024

Received  July 2012 Revised  June 2013 Published  October 2013

In this paper, by using the moving plane method in integral forms, we establish a Liouville type theorem for a coupled integral system with Navier boundary values in the half-space. Furthermore, we prove that the Liouville type theorem is valid for the related differential system as well under an additional assumption by showing the equivalence between the involved differential and integral systems.
Keywords: Navier boundary value, Liouville type theorem, Kelvin transformation., Rotational symmetry, Moving plane method in integral forms.
Mathematics Subject Classification: Primary: 35B53; 35J40; Secondary: 35J6.
Citation: 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
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show all references

References:
[1]

Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247. doi: 10.1017/S0308210500027293.  Google Scholar

[2]

Math. Scand., 2 (1985), 293-345.  Google Scholar

[3]

Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.  Google Scholar

[4]

J. Math. Anal. Appl., 389 (2012), 1365-1373. doi: 10.1016/j.jmaa.2012.01.015.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., suppl. (2005), 164-172.  Google Scholar

[6]

Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[7]

Commun. Pure Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

[8]

Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[9]

Nonlinear Anal., 73 (2010), 2410-2425. doi: 10.1016/j.na.2010.06.012.  Google Scholar

[10]

Ann. Probab., 24 (1996), 293-319. doi: 10.1214/aop/1042644718.  Google Scholar

[11]

Ann. Sc. Norm. Super. Pisa Cl. Sci., 1 (2002), 387-434.  Google Scholar

[12]

Calc. Var. Partial Differential Equations, 15 (2002), 493-517. doi: 10.1007/s005260100134.  Google Scholar

[13]

Adv. Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[14]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1994), 387-397.  Google Scholar

[15]

Nonlinear Anal. Real World Appl., 12 (2011), 3515-3530. doi: 10.1016/j.nonrwa.2011.06.012.  Google Scholar

[16]

Proc. Amer. Math. Soc., 134 (2006), 1661-1670 . doi: 10.1090/S0002-9939-05-08411-X.  Google Scholar

[17]

Proc. Amer. Math. Soc., 140 (2012), 541-551. doi: 10.1090/S0002-9939-2011-11401-1.  Google Scholar

[18]

SIAM J. Appl. Math., 43 (1983), 465-475. doi: 10.1137/0143030.  Google Scholar

[19]

SIAM J. Math. Anal., 40 (2008), 1049-1057. doi: 10.1137/080712301.  Google Scholar

[20]

Proc. Amer. Math. Soc., 138 (2010), 2779-2791. doi: 10.1090/S0002-9939-10-10368-2.  Google Scholar

[21]

J. Differential Equations, 225 (2006), 685-709. doi: 10.1016/j.jde.2005.10.016.  Google Scholar

[22]

Commun. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[23]

Differential Integral Equations, 9 (1996), 465-479.  Google Scholar

[24]

J. Differential Equations, 233 (2007), 151-180. doi: 10.1016/j.jde.2006.09.018.  Google Scholar

[25]

in "Handbook of Differential Equations, Stationary Partial Differential Equations," Volume 3, Chapter 7 (Eds. M. Chipot and P. Quittner), Elsevier, 2006. Google Scholar

[26]

Birkhauser Boston, Inc., Boston, MA, 2001.  Google Scholar

[27]

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[28]

Math. Z., 261 (2009), 805-827. doi: 10.1007/s00209-008-0352-3.  Google Scholar

[29]

J. Differential Equations, 248 (2010), 1866-1878. doi: 10.1016/j.jde.2009.09.012.  Google Scholar

[30]

Differential Integral Equations, 9 (1996), 635-653.  Google Scholar

[31]

J. Math. Pures Appl., 89 (2008), 114-133. doi: 10.1016/j.matpur.2007.10.003.  Google Scholar

[32]

J. Differential Equations, 161 (2000), 110-153. doi: 10.1006/jdeq.1999.3698.  Google Scholar

[33]

Math. Ann., 313 (1999), 207-228. doi: 10.1017/S0308210500027293.  Google Scholar

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