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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
References:
[1]

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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.  Google Scholar

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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.  Google Scholar

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W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Equations, 9 (1996), 465-479.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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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.  Google Scholar

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M. Zhu, Liouville theorems on some indefinite equations, Proc. Roy. Soc. Edinburgh Sect. A Math., 129 (1999), 649-661. doi: 10.1017/S0308210500021569.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[11]

M. Lazzo and P. Schmidt, Nonexistence criteria for polyharmonic boundary-value problems, Analysis, 28 (2008), 449-460. doi: 10.1524/anly.2008.0928.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$, Differential & Integral Equations, 9 (1996), 465-479.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

M. Zhu, Liouville theorems on some indefinite equations, Proc. Roy. Soc. Edinburgh Sect. A Math., 129 (1999), 649-661. doi: 10.1017/S0308210500021569.  Google Scholar

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