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March  2016, 36(3): 1721-1736. doi: 10.3934/dcds.2016.36.1721

A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$

Lizhi Zhang 1, , Congming Li 2, , Wenxiong Chen 3, and  Tingzhi Cheng 4,

1. 

School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
2. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240
3. 

Department of Mathematics, Yeshiva University, New York, NY 10033
4. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

Received  November 2014 Revised  April 2015 Published  August 2015

In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2} u(x)=0,~u(x)\geq0, & \qquad x\in\mathbb{R}^n_+, \\ u(x)\equiv0, & \qquad x\notin\mathbb{R}^{n}_{+}. \end{array}\right.                      (1) \end{equation} We prove that all solutions of (1) are either identically zero or assuming the form \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2} \end{equation} for some positive constant $C$.
Keywords: Poisson representation., $\alpha$-harmonic functions, The fractional Laplacian, uniqueness of solutions, Liouville theorem.
Mathematics Subject Classification: Primary: 35S05, 35B53, 35B0.
Citation: Lizhi Zhang, Congming Li, Wenxiong Chen, Tingzhi Cheng. A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1721-1736. doi: 10.3934/dcds.2016.36.1721
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show all references

References:
[1]

Illinois J. Math., 46 (2002), 541-556.  Google Scholar

[2]

Milan J. Math., 76 (2008), 27-67. doi: 10.1007/s00032-008-0090-3.  Google Scholar

[3]

Nonlinear Anal., 121 (2015), 370-381. doi: 10.1016/j.na.2014.11.003.  Google Scholar

[4]

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

[5]

Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

[6]

Contemp. Math., 528 (2010), 105-114. doi: 10.1090/conm/528/10417.  Google Scholar

[7]

M. Fall, Entire s-harmonic functions are affine,, preprint, ().   Google Scholar

[8]

M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space,, preprint, ().   Google Scholar

[9]

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

[10]

Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag Berlin Heidelberg, New York, 1972. doi: 10.1007/978-3-642-65183-0.  Google Scholar

[11]

Analysis, 28 (2008), 449-460. doi: 10.1524/anly.2008.0928.  Google Scholar

[12]

J. Differential Equations, 247 (2009), 1479-1504. doi: 10.1016/j.jde.2009.05.005.  Google Scholar

[13]

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

[14]

Nonlinear Analysis, 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.  Google Scholar

[15]

Pacific J. Math., 253 (2011), 455-473. doi: 10.2140/pjm.2011.253.455.  Google Scholar

[16]

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

[17]

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

[18]

Comm. Pure Appl. Math., 38 (1985), 679-684. doi: 10.1002/cpa.3160380515.  Google Scholar

[19]

J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[20]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations,, preprint, ().   Google Scholar

[21]

Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[22]

Proc. Roy. Soc. Edinburgh Sect. A Math., 129 (1999), 649-661. doi: 10.1017/S0308210500021569.  Google Scholar

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