# American Institute of Mathematical Sciences

March  2019, 39(3): 1559-1571. doi: 10.3934/dcds.2019068

## Symmetry properties in systems of fractional Laplacian equations

 1 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 2 Department of Applied Mathematics, University of Colorado Boulder, Boulder, CO 80302, USA

* Corresponding author: Hao Xu

Received  April 2018 Revised  May 2018 Published  December 2018

Fund Project: The first author is supported by Natural Science Foundation of Shanghai grant 16ZR1402100

We consider the systems of fractional Laplacian equations in a domain(bounded or unbounded) in $\mathbb{R}^n$. By using a direct method of moving planes, we show that $u_i(x)$ ($i = 1,2,···,m$) are radial symmetric about the same point and strictly decreasing in the radial direction with respect to this point. Comparing with Zhuo-Chen-Cui-Yuan [38], our results not only include subcritical case and critical case but also include supercritical case, and we need not the nonlinear terms to be homogenous. In addition, we completely remove the nonnegativity of $\frac{\partial f_i}{\partial u_i}$. Above all, to the best of our knowledge, it is the first result on the symmetric property of the system containing the gradient of the solution in the nonlinear terms.

Citation: Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068
##### References:
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Differential Equations, 163 (2000), 41-56. doi: 10.1006/jdeq.1999.3701. Google Scholar [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar [7] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Mathematics, 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [9] M. Cai and L. Ma, Moving planes for nonlinear fractional Lapla- cian equation with negative powers, Discrete and Continuous Dynamical Systems-Series A, 38 (2018), 4603-4615. doi: 10.3934/dcds.2018201. Google Scholar [10] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Communications in Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. Google Scholar [11] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. Google Scholar [12] 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 [13] W. Chen and C. Li, Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar [14] W. Chen, C. Li and B. 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G. de Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123. doi: 10.1007/BF01193947. Google Scholar [19] D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 21 (1994), 387-397. Google Scholar [20] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Communications in Mathematical Physics, 68 (1979), 209-243. doi: 10.1007/BF01221125. Google Scholar [21] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Adv. Math. Suppl. Stud. A, 7 (1981), 369-402. Google Scholar [22] S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Matematica Pura ed Applicata, 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y. Google Scholar [23] C. Li, Z. G. Wu and H. Xu, Maximum principles and Bôcher type theorems, Pro Natl Acad Sci USA, 115 (2018), 6976-6979. Google Scholar [24] C. Li and Z. G. Wu, Radial symmetry for systems of fractional Laplacian, Acta Mathematica Scientia, 38 (2018), 1567-1582. doi: 10.1016/S0252-9602(18)30832-4. Google Scholar [25] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Communications in Partial Differential Equations, 16 (1991), 491-526. doi: 10.1080/03605309108820766. Google Scholar [26] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Communications in Partial Differential Equations, 16 (1991), 585-615. doi: 10.1080/03605309108820770. Google Scholar [27] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Analysis: Theory, Methods Applications, 146 (2016), 120-135. doi: 10.1016/j.na.2016.08.022. Google Scholar [28] B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Discrete and Continuous Dynamical Systems - Series A, 38 (2018), 5339-5349. doi: 10.3934/dcds.2018235. Google Scholar [29] Y. Lü and C. Zhou, Symmetry for an integral system with general nonlinearity, Discrete Continuous Dynamical Systems -A, 39 (2019), 1533-1543. doi: 10.3934/dcds.2018121. Google Scholar [30] L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Advances in Mathematics, 225 (2010), 3052-3063. doi: 10.1016/j.aim.2010.05.022. Google Scholar [31] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [32] J. Serrin, A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, 43 (1971), 304-318. doi: 10.1007/BF00250468. Google Scholar [33] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, Journal Mathematical Analysis Applications, 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar [34] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications Pure Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar [35] B. Sirakov, On symmetry in elliptic systems, Appl. Anal., 41 (1991), 1-9. doi: 10.1080/00036819108840012. Google Scholar [36] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. doi: 10.1016/0022-0396(81)90113-3. Google Scholar [37] L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $α$-harmonic functions in $\mathbb{R}_+^n$, Discrete Continuous Dynamical Systems -A, 36 (2016), 1721-1736. doi: 10.3934/dcds.2016.36.1721. Google Scholar [38] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Discrete Continuous Dynamical Systems -A, 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125. Google Scholar

show all references

##### References:
 [1] A. D. Aleksandrov, Uniqueness theorems for surfaces in the large, Amer. Math. Soc. Transl., 21 (1962), 412-416. Google Scholar [2] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, Journal of Geometry and Physics, 5 (1988), 237-275. doi: 10.1016/0393-0440(88)90006-X. Google Scholar [3] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Boletim da Sociedade Brasileira de Matemática-Bulletin/Brazilian Mathematical Society, 22 (1991), 1-37. doi: 10.1007/BF01244896. Google Scholar [4] C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concaveconvex elliptic problem involving the fractional Laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. Google Scholar [5] J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differential Equations, 163 (2000), 41-56. doi: 10.1006/jdeq.1999.3701. Google Scholar [6] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar [7] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Mathematics, 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025. Google Scholar [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar [9] M. Cai and L. Ma, Moving planes for nonlinear fractional Lapla- cian equation with negative powers, Discrete and Continuous Dynamical Systems-Series A, 38 (2018), 4603-4615. doi: 10.3934/dcds.2018201. Google Scholar [10] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Communications in Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954. Google Scholar [11] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. Google Scholar [12] 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 [13] W. Chen and C. Li, Classifcation of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Mathematica Scientia, 29 (2009), 949-960. doi: 10.1016/S0252-9602(09)60079-5. Google Scholar [14] W. Chen, C. Li and B. Ou, Classifcation of solutions for an integral equation, Communications on Pure and Applied Mathematics, 59 (2006), 330-343. doi: 10.1002/cpa.20116. Google Scholar [15] T. Cheng, G. Huang and C. Li, The maximum principles for fractional Laplacian equations and their applications, Communications in Contemporary Mathematics, 19 (2017), 1750018(12pages). doi: 10.1142/S0219199717500183. Google Scholar [16] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216. Google Scholar [17] P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math., 16 (2014), 1350023(24pages). doi: 10.1142/S0219199713500235. Google Scholar [18] D. G. de Figueiredo, Monotonicity and symmetry of solutions of elliptic systems in general domains, NoDEA Nonlinear Differential Equations Appl., 1 (1994), 119-123. doi: 10.1007/BF01193947. Google Scholar [19] D. G. de Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 21 (1994), 387-397. Google Scholar [20] B. Gidas, W. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Communications in Mathematical Physics, 68 (1979), 209-243. doi: 10.1007/BF01221125. Google Scholar [21] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Adv. Math. Suppl. Stud. A, 7 (1981), 369-402. Google Scholar [22] S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Annali di Matematica Pura ed Applicata, 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y. Google Scholar [23] C. Li, Z. G. Wu and H. Xu, Maximum principles and Bôcher type theorems, Pro Natl Acad Sci USA, 115 (2018), 6976-6979. Google Scholar [24] C. Li and Z. G. Wu, Radial symmetry for systems of fractional Laplacian, Acta Mathematica Scientia, 38 (2018), 1567-1582. doi: 10.1016/S0252-9602(18)30832-4. Google Scholar [25] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Communications in Partial Differential Equations, 16 (1991), 491-526. doi: 10.1080/03605309108820766. Google Scholar [26] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Communications in Partial Differential Equations, 16 (1991), 585-615. doi: 10.1080/03605309108820770. Google Scholar [27] B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Analysis: Theory, Methods Applications, 146 (2016), 120-135. doi: 10.1016/j.na.2016.08.022. Google Scholar [28] B. Liu, Direct method of moving planes for logarithmic Laplacian system in bounded domains, Discrete and Continuous Dynamical Systems - Series A, 38 (2018), 5339-5349. doi: 10.3934/dcds.2018235. Google Scholar [29] Y. Lü and C. Zhou, Symmetry for an integral system with general nonlinearity, Discrete Continuous Dynamical Systems -A, 39 (2019), 1533-1543. doi: 10.3934/dcds.2018121. Google Scholar [30] L. Ma and B. Liu, Symmetry results for decay solutions of elliptic systems in the whole space, Advances in Mathematics, 225 (2010), 3052-3063. doi: 10.1016/j.aim.2010.05.022. Google Scholar [31] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées, 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar [32] J. Serrin, A symmetry problem in potential theory, Archive for Rational Mechanics and Analysis, 43 (1971), 304-318. doi: 10.1007/BF00250468. Google Scholar [33] R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, Journal Mathematical Analysis Applications, 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar [34] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Communications Pure Applied Mathematics, 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar [35] B. Sirakov, On symmetry in elliptic systems, Appl. Anal., 41 (1991), 1-9. doi: 10.1080/00036819108840012. Google Scholar [36] W. C. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations, 42 (1981), 400-413. doi: 10.1016/0022-0396(81)90113-3. Google Scholar [37] L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $α$-harmonic functions in $\mathbb{R}_+^n$, Discrete Continuous Dynamical Systems -A, 36 (2016), 1721-1736. doi: 10.3934/dcds.2016.36.1721. Google Scholar [38] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Discrete Continuous Dynamical Systems -A, 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125. Google Scholar
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