# American Institute of Mathematical Sciences

August  2017, 37(7): 3587-3599. doi: 10.3934/dcds.2017154

## Monotonicity and symmetry of solutions to fractional Laplacian equation

 School of Mathematical Sciences, Shanghai Jiaotong University, Shanghai 200240, China

* Corresponding author: Tingzhi Cheng

Received  December 2015 Revised  February 2017 Published  April 2017

Fund Project: The author is supported by Research Grant for Graduate Students of Shanghai Jiaotong University 201701.

Let
 $0 < \alpha < 2$
be any real number and let
 $\Omega$
be an open domain in
 $\mathbb R^{n}$
. Consider the following Dirichlet problem of a semi-linear equation involving the fractional Laplacian:
 $$$\left\{\begin{array}{ll}(-\Delta)^{\alpha/2} u(x)=f(x,u,\nabla{u}),~u(x)>0,&\qquad x\in{\Omega}, \\u(x)\equiv0,&\qquad x\notin{\Omega}.\end{array}\right. \tag{1}\label{p1}$$$
In this paper, instead of using the conventional extension method introduced by Caffarelli and Silvestre, we employ a direct method of moving planes for the fractional Laplacian to obtain the monotonicity and symmetry of the positive solutions of a semi-linear equation involving the fractional Laplacian. By using the integral definition of the fractional Laplacian, we first introduce various maximum principles which play an important role in the process of moving planes. Then we establish the monotonicity and symmetry of positive solutions of the semi-linear equations involving the fractional Laplacian.
Citation: Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154
##### References:
 [1] F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Diff. Equ., 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2. [2] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a halfspace, in Boundary Value Problems for Partial Differential Equations and Applications (eds. volume dedicated to E. Magenes, J. L. Lions et al. ), Masson, Paris, 29 (1993), 27-42. [3] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities 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. [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] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. and Physics, 5 (1988), 237-275.  doi: 10.1016/0393-0440(88)90006-X. [7] C. Brandle, E. Colorado, de Pablo A. and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [8] H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth. Partial differential equations and the calculus of variations, Vol. Ⅰ, Progr, Nonlinear Differential Equations Appl.. Birkh"auser Boston. Boston, MA, 1 (1989), 149-192. [9] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [10] 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. [11] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [12] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8. [13] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst, 4 (2010), xii+299 pp. [14] 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. [15] W. Chen, C. Li and B. Ou, Qualitative properities of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347. [16] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038. [17] C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684. [18] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Advances in Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018. [19] D. G. Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. [20] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [21] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, in Math. Anal. and Applications, Part A, Advances in Math. Suppl. (ed. L. Nachbin), Academic Pr. , Studies, 7 (1981), 369-402. [22] H. G. Kaper and M. K. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17.  doi: 10.1007/978-1-4613-9608-6_1. [23] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^n$, Archive for Rational Mechanics and Analysis, 105 (19689), 243-266.  doi: 10.1007/BF00251502. [24] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. in Partial Differential Equations, 16 (1991), 491-526.  doi: 10.1080/03605309108820766. [25] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. in Partial Differential Equations, 16 (1991), 585-615.  doi: 10.1080/03605309108820770. [26] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874. [27] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651. [28] L. Zhang and T. Cheng, Liouville theorems involving the fractional Laplacian on the upper half Euclidean space, submitted to Acta Applicandae Mathematicae.

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##### References:
 [1] F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Diff. Equ., 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2. [2] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a halfspace, in Boundary Value Problems for Partial Differential Equations and Applications (eds. volume dedicated to E. Magenes, J. L. Lions et al. ), Masson, Paris, 29 (1993), 27-42. [3] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities 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. [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] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. and Physics, 5 (1988), 237-275.  doi: 10.1016/0393-0440(88)90006-X. [7] C. Brandle, E. Colorado, de Pablo A. and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175. [8] H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth. Partial differential equations and the calculus of variations, Vol. Ⅰ, Progr, Nonlinear Differential Equations Appl.. Birkh"auser Boston. Boston, MA, 1 (1989), 149-192. [9] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025. [10] 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. [11] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198.  doi: 10.1016/j.aim.2014.12.013. [12] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8. [13] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS. Ser. Differ. Equ. Dyn. Syst, 4 (2010), xii+299 pp. [14] 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. [15] W. Chen, C. Li and B. Ou, Qualitative properities of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347. [16] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437.  doi: 10.1016/j.aim.2016.11.038. [17] C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684. [18] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Advances in Math., 229 (2012), 2835-2867.  doi: 10.1016/j.aim.2012.01.018. [19] D. G. Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. [20] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125. [21] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, in Math. Anal. and Applications, Part A, Advances in Math. Suppl. (ed. L. Nachbin), Academic Pr. , Studies, 7 (1981), 369-402. [22] H. G. Kaper and M. K. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States Ⅱ, 13 (1988), 1-17.  doi: 10.1007/978-1-4613-9608-6_1. [23] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^n$, Archive for Rational Mechanics and Analysis, 105 (19689), 243-266.  doi: 10.1007/BF00251502. [24] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. in Partial Differential Equations, 16 (1991), 491-526.  doi: 10.1080/03605309108820766. [25] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. in Partial Differential Equations, 16 (1991), 585-615.  doi: 10.1080/03605309108820770. [26] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874. [27] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651. [28] L. Zhang and T. Cheng, Liouville theorems involving the fractional Laplacian on the upper half Euclidean space, submitted to Acta Applicandae Mathematicae.
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