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Liouville type theorems for fractional and higher-order fractional systems
Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators
1. | School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China |
2. | School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
$ A_{\alpha} u(x) = C_{n,\alpha} \rm{P.V.} \int_{\mathbb{R}^n} \frac{a(x-y)(u(x)-u(y))}{|x-y|^{n+\alpha}} dy, $ |
$ a(x) $ |
$ A_{\alpha} $ |
$ \mathbb R^n_+ $ |
References:
[1] |
H. Berestycki, L. A. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris, 29 (1993), 27-42. |
[2] |
H. Berestycki, F. Hamel and R. Monneau,
One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: 10.1215/S0012-7094-00-10331-6. |
[3] |
H. Berestycki and L. Nirenberg,
Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[4] |
H. Berestycki and L. Nirenberg, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, in Analysis, et Cetera, Academic Press, Boston, MA, (1990), 115-164. |
[5] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[8] |
L. Caffarelli and L. Silvestre,
Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
doi: 10.1007/s00205-010-0336-4. |
[9] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[10] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[11] |
W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 29, 18 pp.
doi: 10.1007/s00526-017-1110-3. |
[12] |
W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math., 27 (2016), 1650064, 20 pp.
doi: 10.1142/S0129167X16500646. |
[13] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[14] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[15] |
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. |
[16] |
W. Chen and S. Qi,
Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310.
doi: 10.3934/dcds.2019055. |
[17] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[18] |
X. Chen, G. Bao and G. Li, The sliding method for the nonlocal Monge-Ampère operator, Nonlinear Anal., 196 (2020), 111786, 13 pp.
doi: 10.1016/j.na.2020.111786. |
[19] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), 1750018, 12.
doi: 10.1142/S0219199717500183. |
[20] |
C. Li, Z. Wu and H. Xu,
Maximum principles and Bôcher type theorems, Proc. Natl. Acad. Sci. USA, 115 (2018), 6976-6979.
doi: 10.1073/pnas.1804225115. |
[21] |
Z. Liu, Maximum principles and monotonicity of solutions for fractional $p$-equations in unbounded domains, J. Differential Equations, 270 (2021), 1043-1078. arXiv: 1905.06493.
doi: 10.1016/j.jde.2020.09.001. |
[22] |
L. Ma and Z. Zhang, Monotonicity of positive solutions for fractional $p$-systems in unbounded Lipschitz domains, Nonlinear Anal., 198 (2020), 111892, 18 pp.
doi: 10.1016/j.na.2020.111892. |
[23] |
D. Tang,
Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian, Math. Methods Appl. Sci., 40 (2017), 2596-2609.
doi: 10.1002/mma.4184. |
[24] |
L. Wu and W. Chen, Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities. (in chinese), Sci. Sin. Math., (2020), to appear. Google Scholar |
[25] |
L. Wu and W. Chen, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 106933, 26 pp.
doi: 10.1016/j.aim.2019.106933. |
show all references
References:
[1] |
H. Berestycki, L. A. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a half space, in Boundary Value Problems for Partial Differential Equations and Applications, RMA Res. Notes Appl. Math., Masson, Paris, 29 (1993), 27-42. |
[2] |
H. Berestycki, F. Hamel and R. Monneau,
One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: 10.1215/S0012-7094-00-10331-6. |
[3] |
H. Berestycki and L. Nirenberg,
Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[4] |
H. Berestycki and L. Nirenberg, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, in Analysis, et Cetera, Academic Press, Boston, MA, (1990), 115-164. |
[5] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[8] |
L. Caffarelli and L. Silvestre,
Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.
doi: 10.1007/s00205-010-0336-4. |
[9] |
W. Chen, Y. Fang and R. Yang,
Liouville theorems involving the fractional Laplacian on a half space, Adv. Math., 274 (2015), 167-198.
doi: 10.1016/j.aim.2014.12.013. |
[10] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[11] |
W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 29, 18 pp.
doi: 10.1007/s00526-017-1110-3. |
[12] |
W. Chen, C. Li and Y. Li, A direct blowing-up and rescaling argument on nonlocal elliptic equations, Internat. J. Math., 27 (2016), 1650064, 20 pp.
doi: 10.1142/S0129167X16500646. |
[13] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[14] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[15] |
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. |
[16] |
W. Chen and S. Qi,
Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310.
doi: 10.3934/dcds.2019055. |
[17] |
W. Chen and J. Zhu,
Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260 (2016), 4758-4785.
doi: 10.1016/j.jde.2015.11.029. |
[18] |
X. Chen, G. Bao and G. Li, The sliding method for the nonlocal Monge-Ampère operator, Nonlinear Anal., 196 (2020), 111786, 13 pp.
doi: 10.1016/j.na.2020.111786. |
[19] |
T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math., 19 (2017), 1750018, 12.
doi: 10.1142/S0219199717500183. |
[20] |
C. Li, Z. Wu and H. Xu,
Maximum principles and Bôcher type theorems, Proc. Natl. Acad. Sci. USA, 115 (2018), 6976-6979.
doi: 10.1073/pnas.1804225115. |
[21] |
Z. Liu, Maximum principles and monotonicity of solutions for fractional $p$-equations in unbounded domains, J. Differential Equations, 270 (2021), 1043-1078. arXiv: 1905.06493.
doi: 10.1016/j.jde.2020.09.001. |
[22] |
L. Ma and Z. Zhang, Monotonicity of positive solutions for fractional $p$-systems in unbounded Lipschitz domains, Nonlinear Anal., 198 (2020), 111892, 18 pp.
doi: 10.1016/j.na.2020.111892. |
[23] |
D. Tang,
Positive solutions to semilinear elliptic equations involving a weighted fractional Lapalacian, Math. Methods Appl. Sci., 40 (2017), 2596-2609.
doi: 10.1002/mma.4184. |
[24] |
L. Wu and W. Chen, Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities. (in chinese), Sci. Sin. Math., (2020), to appear. Google Scholar |
[25] |
L. Wu and W. Chen, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 106933, 26 pp.
doi: 10.1016/j.aim.2019.106933. |
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