November  2020, 19(11): 5269-5283. doi: 10.3934/cpaa.2020237

Monotonicity of solutions for a class of nonlocal Monge-Ampère problem

School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical, Sciences, Central China Normal University, Wuhan, 430079, China, Department of Mathematical Sciences, Yeshiva University, New York, NY, USA

Received  April 2020 Revised  June 2020 Published  November 2020 Early access  September 2020

Fund Project: This work was supported by Natural Science Foundation of China (Grant No.11771166), Hubei Key Laboratory of Mathematical Sciences, Program for Changjiang Scholars and Innovative Research Team in University #IRT_17R46 and China Scholarship Council

In this paper, we consider nonlinear problems involving nonlocal Monge-Ampère operators. By using a sliding method, we establish monotonicity of positive solutions for nonlocal Monge-Ampère problems both in an infinite slab and in an upper half space. During this process, an important idea we applied is to estimate the singular integrals defining the nonlocal Monge-Ampère operator along a sequence of approximate maximum points. It allows us to assume weaker conditions on nonlinear terms. Another idea is to employ a generalized average inequality which plays an important role and greatly simplify the process of the sliding.

Citation: Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237
References:
[1]

H. BerestyckiL. Caffarelli and L. Nirenberg, Inequalitites for second-order elliptic equations with applications to unbounded domains, I, Duke Math. J., 81 (1996), 467-494.  doi: 10.1215/S0012-7094-96-08117-X.

[2]

H. BerestyckiL. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Commun. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.

[3]

H. Berestycki and L. Nirenberg, Monotonicity, symmetry and anti- symmetry 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 semi-linear elliptic equations in cylindrical domains, 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 F. Charro, On a fractional Monge-Ampère operator, Ann. PDE., 1 (2015), 47pp. doi: 10.1007/s40818-015-0005-x.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. Caffarelli and L. Silvestre, A nonlocal Monge-Ampère equation, Commun. Anal. Geom., 24 (2016), 307-335.  doi: 10.4310/CAG.2016.v24.n2.a4.

[9]

W. Chen and C. Li, Maximum principle for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[10]

W. ChenC. Li and 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.

[11]

W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019.

[12]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[13]

W. Chen and S. Qi, Direct methods on fractional equations, Disc. Cont. Dyna. Sys., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.

[14]

S. DipierroN. Soave and E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.  doi: 10.1007/s00208-016-1487-x.

[15]

Z. Liu and W. Chen, Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains, arXiv: 1905.06493.

[16]

L. Wu and W. Chen, The sliding method for the fractional p-Laplacian, Adv. Math., 361 (2020), 106933. doi: 10.1016/j.aim.2019.106933.

[17]

L. Wu and W. Chen, Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities, arXiv: 1905.09999.

show all references

References:
[1]

H. BerestyckiL. Caffarelli and L. Nirenberg, Inequalitites for second-order elliptic equations with applications to unbounded domains, I, Duke Math. J., 81 (1996), 467-494.  doi: 10.1215/S0012-7094-96-08117-X.

[2]

H. BerestyckiL. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Commun. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6.

[3]

H. Berestycki and L. Nirenberg, Monotonicity, symmetry and anti- symmetry 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 semi-linear elliptic equations in cylindrical domains, 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 F. Charro, On a fractional Monge-Ampère operator, Ann. PDE., 1 (2015), 47pp. doi: 10.1007/s40818-015-0005-x.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. PDE., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[8]

L. Caffarelli and L. Silvestre, A nonlocal Monge-Ampère equation, Commun. Anal. Geom., 24 (2016), 307-335.  doi: 10.4310/CAG.2016.v24.n2.a4.

[9]

W. Chen and C. Li, Maximum principle for the fractional p-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.  doi: 10.1016/j.aim.2018.07.016.

[10]

W. ChenC. Li and 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.

[11]

W. Chen, C. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, 2019.

[12]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[13]

W. Chen and S. Qi, Direct methods on fractional equations, Disc. Cont. Dyna. Sys., 39 (2019), 1269-1310.  doi: 10.3934/dcds.2019055.

[14]

S. DipierroN. Soave and E. Valdinoci, On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.  doi: 10.1007/s00208-016-1487-x.

[15]

Z. Liu and W. Chen, Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains, arXiv: 1905.06493.

[16]

L. Wu and W. Chen, The sliding method for the fractional p-Laplacian, Adv. Math., 361 (2020), 106933. doi: 10.1016/j.aim.2019.106933.

[17]

L. Wu and W. Chen, Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities, arXiv: 1905.09999.

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