# American Institute of Mathematical Sciences

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

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

 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 $\mathbb{R}^n_+$: $$\left\{\begin{array}{ll} (-\triangle)^{\alpha/2} u(x)=0,~u(x)\geq0, & \qquad x\in\mathbb{R}^n_+, \\ u(x)\equiv0, & \qquad x\notin\mathbb{R}^{n}_{+}. \end{array}\right. (1)$$ We prove that all solutions of (1) are either identically zero or assuming the form $$u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+}, \end{array}\right. \label{2}$$ for some positive constant $C$.
Citation: Lizhi Zhang, Congming Li, Wenxiong Chen, Tingzhi Cheng. A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1721-1736. doi: 10.3934/dcds.2016.36.1721
##### References:
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##### References:
 [1] K. Bogdan, T. Kulczycki and A. Nowak, Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes, Illinois J. Math., 46 (2002), 541-556. [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: 10.1007/s00032-008-0090-3. [3] W. Chen, L. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381. doi: 10.1016/j.na.2014.11.003. [4] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [5] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354. [6] L. Dupaigne and Y. Sire, A Liouville theorem for non-local elliptic equations, Symmetry for elliptic PDEs, Contemp. Math., 528 (2010), 105-114. doi: 10.1090/conm/528/10417. [7] M. Fall, Entire s-harmonic functions are affine, preprint, arXiv:1407.5934. [8] M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space, preprint, arXiv:1309.7230. [9] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problem in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [10] N. S. Landkof, Foundations of Modern Potential Theory, 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. [11] M. Lazzo and P. Schmidt, Nonexistence criteria for polyharmonic boundary-value problems, Analysis, 28 (2008), 449-460. doi: 10.1524/anly.2008.0928. [12] M. Lazzo and P. Schmidt, Oscillatory radial solutions for subcritical biharmonic equations, J. Differential Equations, 247 (2009), 1479-1504. doi: 10.1016/j.jde.2009.05.005. [13] Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8. [14] G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis, 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036. [15] G. Lu and J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math., 253 (2011), 455-473. doi: 10.2140/pjm.2011.253.455. [16] L. Ma and D. Chen, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [17] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Equations, 9 (1996), 465-479. [18] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math., 38 (1985), 679-684. doi: 10.1002/cpa.3160380515. [19] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. [20] X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, preprint, arXiv:1404.1197. [21] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [22] M. Zhu, Liouville theorems on some indefinite equations, Proc. Roy. Soc. Edinburgh Sect. A Math., 129 (1999), 649-661. doi: 10.1017/S0308210500021569.
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