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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 |
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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