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Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation

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

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  • 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_+ $.

    Mathematics Subject Classification: Primary: 35R11; Secondary: 35B33.

    Citation:

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  • [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
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