# 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 & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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. Google Scholar [18] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, Comm. Pure Appl. Math., 38 (1985), 679. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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. Google Scholar [18] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, Comm. Pure Appl. Math., 38 (1985), 679. 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. 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. 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. doi: 10.1017/S0308210500021569. Google Scholar
 [1] Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555 [2] Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248 [3] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [4] Seppo Granlund, Niko Marola. Phragmén--Lindelöf theorem for infinity harmonic functions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 127-132. doi: 10.3934/cpaa.2015.14.127 [5] Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113 [6] Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865 [7] Ammari Zied, Liard Quentin. On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 723-748. doi: 10.3934/dcds.2018032 [8] Laura Abatangelo, Susanna Terracini. Harmonic functions in union of chambers. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5609-5629. doi: 10.3934/dcds.2015.35.5609 [9] De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431 [10] Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393 [11] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [12] Serena Dipierro, Enrico Valdinoci. On a fractional harmonic replacement. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3377-3392. doi: 10.3934/dcds.2015.35.3377 [13] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813 [14] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71 [15] Kunquan Lan, Wei Lin. Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 849-861. doi: 10.3934/dcdsb.2016.21.849 [16] Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397 [17] Anton Petrunin. Harmonic functions on Alexandrov spaces and their applications. Electronic Research Announcements, 2003, 9: 135-141. [18] Dag Lukkassen, Annette Meidell, Peter Wall. On the conjugate of periodic piecewise harmonic functions. Networks & Heterogeneous Media, 2008, 3 (3) : 633-646. doi: 10.3934/nhm.2008.3.633 [19] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [20] Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003

2018 Impact Factor: 1.143