# American Institute of Mathematical Sciences

May  2021, 41(5): 2285-2300. 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  May 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (5) : 2285-2300. doi: 10.3934/dcds.2020362
##### 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. [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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##### 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. [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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