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November  2014, 34(11): 4947-4966. doi: 10.3934/dcds.2014.34.4947

Liouville type theorem for nonlinear elliptic equation with general nonlinearity

1. 

The Center for China's Overseas Interests, Shenzhen University, Shenzhen Guangdong, 518060, China

Received  September 2013 Revised  December 2013 Published  May 2014

In this paper, we study the nonexistence of positive solutions for the following elliptic equation $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u=f(u) & in \quad \mathbb{R}_+^N, \displaystyle \\ \frac{\partial u}{\partial \nu}=g(u) & on \quad \partial \mathbb{R}_+^N \end{array} \right. $$ and elliptic system $$ \left\{ \begin{array}{ll} \displaystyle -\Delta u_1=f_1(u_1,u_2) &in \quad \mathbb{R}_+^N, \\ \\-\Delta u_2=f_2(u_1,u_2) & in\quad \mathbb{R}_+^N, \\ \displaystyle \\ \frac{\partial u_1}{\partial \nu}=g_1(u_1,u_2),\quad \frac{\partial u_2}{\partial \nu}=g_2(u_1,u_2) & on \quad \partial \mathbb{R}_+^N. \end{array} \right. $$ We will prove that these problems possess no positive solutions under some assumptions on nonlinear terms. The main technique we use is the moving plane method in an integral form.
Citation: Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
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show all references

References:
[1]

Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[2]

Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[3]

Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5.  Google Scholar

[4]

AIMS Book Series, vol. 4, 2010.  Google Scholar

[5]

Discrete Contin. Dyn. Syst., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083.  Google Scholar

[6]

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

[7]

Comm. P.D.E., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[8]

J. Math. Anal. Appl., 223 (1998), 429-471. doi: 10.1006/jmaa.1998.5958.  Google Scholar

[9]

Rev. Mat. Iberoamericana, 20 (2004), 67-86.  Google Scholar

[10]

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

[11]

J. Math. Pures. Appl., 61 (1982), 41-63.  Google Scholar

[12]

Comm. P.D.E., 6 (1981), 883-901. doi: 10.1002/cpa.3160340406.  Google Scholar

[13]

Commun. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar

[14]

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

[15]

Comm. P.D.E., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[16]

Proceedings of the Royal Society of Edinburgh, 138 (2008), 339-359. doi: 10.1017/S0308210506000394.  Google Scholar

[17]

Ann. Inst. H. Poincare Anal. Non Lineaire, 26 (2009), 1-21. doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[18]

Differential Integral Equations, 7 (1994), 301-313.  Google Scholar

[19]

Trans. Amer. Math. Soc., 346 (1994), 117-135. doi: 10.1090/S0002-9947-1994-1270664-3.  Google Scholar

[20]

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

[21]

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

[22]

Journal d'Analyse Mathématique, 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[23]

Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[24]

Differential Integral Equations, 12 (1999), 601-612.  Google Scholar

[25]

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

[26]

Advances in Mathematics, 226 (2011), 2676-2699. doi: 10.1016/j.aim.2010.07.020.  Google Scholar

[27]

Arch. Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3.  Google Scholar

[28]

Diff. Int. Eq., 9 (1996), 465-479.  Google Scholar

[29]

Differential Integral Equations, 9 (1996), 1157-1164.  Google Scholar

[30]

Diff. Int. Eq., 9 (1996), 635-653.  Google Scholar

[31]

Atti Sem. Mat. Fis. Univ. Modena. Sippl., 46 (1998), 369-380. Google Scholar

[32]

Comm. P.D.E., 23 (1998), 577-599. doi: 10.1080/03605309808821356.  Google Scholar

[33]

Advances in Mathematics, 221 (2009), 1409-1427. doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[34]

Diff. Int. Eq., 8 (1995), 1911-1922.  Google Scholar

[35]

Adv. Diff. Eq., 1 (1996), 241-264.  Google Scholar

[36]

Calc. Var., 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

[37]

X. Yu, Liouville Type Theorems for Singular Integral Equations and Integral Systems,, preprint., ().   Google Scholar

[38]

Journal of Differential Equations, 254 (2013), 2173-2182. doi: 10.1016/j.jde.2012.11.021.  Google Scholar

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