    doi: 10.3934/dcds.2020362

## 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

* Corresponding author: Ting Zhang

Received  June 2020 Revised  August 2020 Published  October 2020

In this paper, we consider the uniformly elliptic nonlocal operators
 $A_{\alpha} u(x) = C_{n,\alpha} \rm{P.V.} \int_{\mathbb{R}^n} \frac{a(x-y)(u(x)-u(y))}{|x-y|^{n+\alpha}} dy,$
where
 $a(x)$
is positively uniform bounded satisfying a cylindrical condition. We first establish the narrow region principle in the bounded domain. Then using the sliding method, we obtain the monotonicity of solutions for the semi-linear equation involving
 $A_{\alpha}$
in both the bounded domain and the whole space. In addition, we establish the maximum principle in the unbounded domain and get the non-existence of solutions in the upper half space
 $\mathbb R^n_+$
.
Citation: Meng Qu, Jiayan Wu, Ting Zhang. Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020362
##### References:
  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar

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##### References:
  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  W. Chen and S. Qi, Direct methods on fractional equations, Discrete Contin. Dyn. Syst., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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  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.  Google Scholar
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