March  2020, 19(3): 1337-1349. doi: 10.3934/cpaa.2020065

Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation

1. 

School of mathematics and statistics, Anhui normal university, Wuhu, 241002, China

2. 

School of Information and Mathematics, Yangtze University, Jingzhou 434023, China

* Corresponding author

Received  March 2019 Revised  July 2019 Published  November 2019

Fund Project: The work was supported by National Natural Science Foundation of China (11871096 and 11671308)

In this paper, we consider equations involving the fully nonlinear fractional order operator with homogeneous Dirichlet condition:
$ \begin{cases} F_\alpha(u)(x) = f(x,u,\nabla u) \ \mbox{in} \ \Omega,\\ u>0, \ \mbox{in}\ \Omega; \ u\equiv0, \ \mbox{in}\ \mathbb R^n\backslash\Omega, \end{cases} $
where
$ \Omega $
is a domain(bounded or unbounded) in
$ \mathbb R^n $
which is convex in
$ x_1- $
direction. By using some ideas of maximum principle, we prove that the solution is strictly increasing in
$ x_1- $
direction in the left half of
$ \Omega $
. Symmetry of solution is also proved. Meanwhile we obtain a Liouville type theorem on the half space
$ \mathbb R^n_+ $
.
Citation: Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065
References:
[1]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

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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

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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

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W. ChenY. 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

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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), Art. 29, 18. doi: 10.1007/s00526-017-1110-3.  Google Scholar

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W. ChenC. 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

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W. ChenC. 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

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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

[9]

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

[10]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[11]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.  Google Scholar

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X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.  Google Scholar

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F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[14]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[15]

Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.  Google Scholar

[16]

C. LiZ. Wu and H. Xu, Maximum principles and bôcher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.  Google Scholar

[17]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.  Google Scholar

[18]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.  Google Scholar

[19]

P. NiuL. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fractional Calculus and Applied Analysis, 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030.  Google Scholar

[20]

Y. Wang and J. Wang, The method of moving planes for integral equation in an extremal case, J. Partial Differ. Equ., 29 (2016), 246-254.  doi: 10.4208/jpde.v29.n3.6.  Google Scholar

show all references

References:
[1]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[2]

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

[3]

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

[4]

W. ChenY. 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

[5]

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), Art. 29, 18. doi: 10.1007/s00526-017-1110-3.  Google Scholar

[6]

W. ChenC. 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

[7]

W. ChenC. 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

[8]

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

[9]

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

[10]

Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018.  Google Scholar

[11]

R. L. Frank and E. H. Lieb, Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality, Calc. Var. Partial Differential Equations, 39 (2010), 85-99.  doi: 10.1007/s00526-009-0302-x.  Google Scholar

[12]

X. HanG. Lu and J. Zhu, Characterization of balls in terms of Bessel-potential integral equation, J. Differential Equations, 252 (2012), 1589-1602.  doi: 10.1016/j.jde.2011.07.037.  Google Scholar

[13]

F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.  doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[14]

F. HangX. Wang and X. Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1-21.  doi: 10.1016/j.anihpc.2007.03.006.  Google Scholar

[15]

Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.  doi: 10.1016/j.jde.2012.11.008.  Google Scholar

[16]

C. LiZ. Wu and H. Xu, Maximum principles and bôcher type theorems, Proceedings of the National Academy of Sciences, 115 (2018), 6976-6979.  doi: 10.1073/pnas.1804225115.  Google Scholar

[17]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577.  doi: 10.1007/s00526-011-0398-7.  Google Scholar

[18]

G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.  doi: 10.1016/j.na.2011.11.036.  Google Scholar

[19]

P. NiuL. Wu and X. Ji, Positive solutions to nonlinear systems involving fully nonlinear fractional operators, Fractional Calculus and Applied Analysis, 21 (2018), 552-574.  doi: 10.1515/fca-2018-0030.  Google Scholar

[20]

Y. Wang and J. Wang, The method of moving planes for integral equation in an extremal case, J. Partial Differ. Equ., 29 (2016), 246-254.  doi: 10.4208/jpde.v29.n3.6.  Google Scholar

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