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

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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

Abstract
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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 and Continuous Dynamical Systems, 2016, 36 (3) : 1721-1736. doi: 10.3934/dcds.2016.36.1721

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.

show all references

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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